The Many-Worlds Interpretation (MWI) is very nice in bra-ket notation! Before diving in, what you wrote for the state of Schrödinger's cat as (no decay $\otimes$ alive cat) + (decay $\otimes$ dead cat) is correct...at least as a toy model of both the radiation-induced poisoning mechanism and cats.
A reasonable paraphrasing of the main premise of MWI is that "all quantum operations are just unitary evolution when we consider a closed system." Measurements only seem weird in the Copenhagen picture because the system is open during the measurement! So if we consider both the physical system and the measurement apparatus, we should recover something unitary!
It turns out, this is not an assumption! Every axiomatic formulation of quantum mechanics implies that the operators on the system form a $C^*$-algebra, and that all valid quantum operations (updates to the density matrix) correspond to completely positive (CP) maps. The Stinespring Dilation Theorem was used by Kraus in the 80s to prove that all quantum operations can be represented as a unitary operation on a "dilated" Hilbert space (followed by a projection on the extra degrees of freedom when the operation is "selective," as with a measurement corresponding to a particular outcome). If we associate these "dilated" degrees of freedom with the state of the measurement apparatus, we see that the unitary implied by the axioms is the same as the unitary time-evolution operator that we actually apply to the system and apparatus during the measurement process!
Let's consider atomic qubits for convenience (e.g., trapped ions), using the computational ($Z$) basis where $Z \left| n \right\rangle = (-1)^n \left| n \right\rangle$ for $n=0,1$. A common way to measure $Z$ on an atomic qubit is using fluorescence measurement, which checks if the qubit is in the state $\left| 1 \right\rangle$ or not by shining photons on the atom to stimulate a transition from the atomic state $\left| 1 \right\rangle$ to some atomic unstable state $\left| a \right\rangle$, which promptly decays back to $\left| 1 \right\rangle$ and we detect the photon. If the atom is in the state $\left| 0 \right\rangle$, nothing happens. So the default state of the apparatus denotes the outcome $\left| 0 \right\rangle$, and detecting a photon flips the state of the apparatus to the state $\left| 1 \right\rangle$ denoting the same outcome.
Generalizing slightly, let's write down the unitary capturing measurement of a Pauli $\sigma^\nu$ on a qubit:
$$ \mathsf{M} \, = \, \frac{1}{2} \left( \mathbb{1} + \sigma^\nu \right) \otimes \widetilde{\mathbb{1}} + \frac{1}{2} \left( \mathbb{1} - \sigma^\nu \right) \otimes \widetilde{X}, $$
where tildes denote the operators acting on the apparatus and the post-measurement state $n=0,1$ of the apparatus denotes the eigenvalue $(-1)^n$ of $\sigma^\nu$ that was observed upon measurement.
In the case of fluorescence measurement of $Z = \sigma^z$, if the atom was in the state $\left| 0 \right\rangle$ then nothing happens when photons are shone on it; if it was in the state $\left| 1 \right\rangle$, the incident light stimulates emission, and the detection of the fluorescent photon by the apparatus is captured by the operator $\widetilde{X}$, which flips the apparatus state from $\left| 0 \right\rangle$ to $\left| 1 \right\rangle$, corresponding to a detection or "click."
As an aside, in the case of the fluorescent measurement described above, the actual physical unitary and dilated Hilbert space is more complicated, but is equivalent to the above when we "bin" all states of the apparatus corresponding to the same outcome into the single representative state $\left| 1 \right\rangle$, and send a large number of photons so that we avoid false negatives.
So the measurement of $Z$ on a qubit in the state $\left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle$ is given by
$$\left| \psi \right\rangle \otimes \left|0\right\rangle = \alpha \left|0 \right\rangle \otimes \left|0 \right\rangle + \beta \left|1 \right\rangle \otimes \left|0 \right\rangle \mapsto \alpha \left| 0 0 \right\rangle + \beta \left| 1 1 \right\rangle \,,$$
meaning that the measurement unitary $\mathsf{M}$, which acts as a CNOT gate with the physical qubit the control qubit and the apparatus the target qubit, merely entangles the state of the apparatus into the physical state of the qubit without altering the structure of the former.
Regarding branches. So we can already answer the part of your question about branches. In MWI, there is no "splitting" of the state of the universe (or "branching") upon measurement. If we write the pre-measurement state of the system in the basis of the operator to be measured, as above, we always find that the measurement merely entangles the state of the detector into the physical state without creating new superpositions. Put another way, measurements only reveal branches that are already there. Moreover, the number of branches is equal to the number of outcomes, and the same outcome does not appear in multiple branches!
Regarding decoherence. Decoherence is merely the mechanism by which we perceive only one outcome. Why should we expect it? Well, for the interaction between the system and apparatus to look like a measurement (the unitary $\mathsf{M}$), we must be able to read off the outcome. This means it must be classically accessible, so the state of the apparatus must somehow be classical and thermally stable. This means it is implicitly subject to decoherence, the mechanism that causes quantum systems to appear classical in the first place.
Classical / thermal systems are highly (if not maximally) entangled with each other. So the measurement apparatus is highly entangled with its environment. This entanglement naturally leads to the appearance of mixed states in some classically preferred basis. We cannot observe coherence superpositions in this basis, which is sort of the meaning of being classical.
Basically, superpositions in the preferred basis (corresponding to symmetries) are "unstable" because the chaotic quantum dynamics that entangles classical systems with their environments (via interactions) causes scrambling, which kills off any quantum coherences (hence the term "decoherence"). At least, as far as other classical objects (whose experience is restricted to the preferred basis) are concerned. What this means in practice is that, if you try to prepare a classical system in a superposition of two classically distinct basis states (e.g., corresponding to different charges of an object, different numbers of photons, etc), then decoherence will absorb that superposition into the state of the entire universe very rapidly. As a result, the coherently superposed state will appear to decay (or collapse) into a single state.
So the need to read the outcomes means that the apparatus is classical. That in turn means that we should expect decoherence. For the states reflecting distinct outcomes to be stable under chaotic dynamics, they must correspond to a symmetry. Most commonly, that symmetry is particle number. We only ever observe a particle or no particle, never superpositions of these two. That's why radioactive decay also appears random to us. This also provides a built-in physical mechanism by which a preferred basis emerges, which resolves another common complaint about the decoherence-based picture of collapse. But this is a basic fact from quantum chaos.
The combination of these ingredients leads to collapse. Without these ingredients, measurements would just look like any other entangling interaction, or else the state of the apparatus following the measurement unitary $\mathsf{M}$ would not reliably indicate the "outcome."
Regarding superpositions and Schrödinger's cat. Yes, the post-measurement state is a superposition, as we saw above. Suppose we intend to measure a single-qubit operator $A$. It turns out that this is equivalent to measuring the involutory part $A'$ of $A$, which is of the form $A' =\sum_\nu \hat{a}_\nu \sigma^\nu$, where $\hat{a}$ is a unit vector. The eigenvalues of $A'$ are $\pm 1$ or $(-1)^n$. We can write the physical state before the measurement in the basis $\left| n \right\rangle$ of $A'$ as
$$ \left| \psi_0 \right\rangle = \sum_{n=0,1} c_n \left| n \right\rangle \, ,$$
so that the "dilated" initial state (or "universal wavefunction") is
$$ \left| \Psi_i \right\rangle = \left| \psi_0 \right\rangle_\text{ph} \otimes \left|0 \right\rangle_\text{det} = \sum_{n=0,1} c_n \left| n \right\rangle_\text{ph} \otimes \left| 0 \right\rangle_\text{det} \, ,$$
and applying the measurement unitary $\mathsf{M}$ leads to
$$ \left| \Psi_f \right\rangle = \mathsf{M} \left| \Psi_i \right\rangle = \frac{1}{2} \sum_{m,n=0,1} c_n \left( \mathbb{1} + (-1)^m A'\right) \otimes \widetilde{X}^m \left| n \right\rangle \otimes \left| 0 \right\rangle = \sum_{m,n} c_n \frac{1 + (-1)^{m+n}}{2} \left| n \right\rangle \otimes \left|m \right\rangle,$$
and the terms with $m \neq n$ have weight zero, leaving
$$ \left| \Psi_f \right\rangle = \sum_{n=0,1} c_n \left| n \right\rangle_\text{ph} \otimes \left| n \right\rangle_\text{det},$$
as expected. So the full state after the measurement is of precisely the form you wrote down if $c_0 = c_1 = 1/\sqrt{2}$!
So basically, if I try to prepare a box with Schrödinger's poor cat in it with the radiation-activated poison, the fact that the cat is alreaady maximally entangled with the rest of the universe actually means that there's not much point in putting the cat in the box lol. But suppose for argument's sake that I prepare the cat's state in isolation, and maintain this isolation while I put the cat in the box. As I start the experiment, I allow interactions with the rest of the universe again. Now, in perfect isolation (i.e., without any interactions with the rest of the universe), sure, the cat can be in a superposition of being both alive and dead. But the second we view the cat's state --- or, more accurately, the second any degree of freedom in the entire rest of the universe interacts with any degree of freedom in the cat --- the superposition of the two states "dead" and "alive" is absorbed into the full state of the universe. The universe then appears to contain two classical realities: one in which the cat is alive, and one in which it is dead. But we can experience only one of these realities (this is an observational fact only), and hence see the cat as alive or dead. If we try to prepare a state in which it's both, decoherence just recollapses that state.
Importantly, this is not merely a conjecture, but an understood result of decoherence and the very mechanism for the emergence of classical and thermal physics from microscopic quantum mechanics!
Importantly, chaotic systems (i.e., dynamical systems that thermalize / relax to thermal equilibrium / obey thermodynamics at late times) forget as much information about their initial conditions as possible! But quantum systems can't actually "forget" anything, because unitary evolution is reversible. What happens is that the evolution of the whole universe (a closed system) indeed retains all information, but the information about a local subsystem is smeared out throughout the universe under dynamics with entangling interactions, so that it can no longer be accessed locally. However, information about symmetries cannot be forgotten --- at least, not as easily --- since there are associated charges to conserve! This is also why hydrodynamics is dominated by conserved charges. It is a key feature of chaotic dynamics and relaxation, and it is why we encode outcomes in states corresponding to different symmetries. And why we can't see superpositions of such states!
The Born rule. People like to say that MWI can't explain the Born rule. That's false. It comes from Gleason's theorem, just like usual. Gleason's theorem says that basically, the only things that naturally give rise to a probability distribution are a complete set of projectors, like the ones that define the eigenspaces of a measurable observable. So that's why probabilities are expectation values of projectors. In the Stinespring (dilated) representation above, we can either evaluate those probabilities in terms of the physical projectors in the physical part of the pre-measurement state to get Born's rule, or evaluate them on the detector part of the post-measurement state. But it's the same principle.
Generalizing the above. I'll also mention that this representation of measurements can also be generalized beyond qubits (see: arXiv:2210.07256, e.g.) and to other types of measurements. What I've described above is the "strong" case of "projective measurements", but nearly identical (and much older) ideas apply to continuous, weak, and generalized measurements. They can always be described as unitary time evolution when the state of the detector is included! In all cases, measurements are simply an entangling interaction between the physical system and some measurment apparatus in a particular basis. That basis always corresponds to the eigenbasis of the observable being measured (for the physical system), while the basis for the state of the measurements apparatus corresponds to a symmetry charge (e.g., particle number), which is robust to the chaotic evolution of the apparatus itself. That chaotic evolution decoheres the state I wrote above, e.g., into a mixed state, which can be understood using random unitary ensembles that don't mix states of the apparatus corresponding to different numbers of particles (and thus, different outcomes).
Some other comments on MWI. I should also note that, in the context of measurements of Bell states where we give one qubit to Alice (A) and one to Bob (B), there is no need for collapse or "spooky action at a distance" or anything else with MWI! The two parties simply entangle their respective apparatus into the entangled Bell state without altering its form. If they measure the same operator, they are guaranteed to get the same outcome because entangling into the state guarantees it! There is no need for action at a distance. In fact, the measurement representation above has been used to prove that all quantum dynamics are local because measuring entangled states does not transmit any information on its own. One must always send a classical signal. If Alice tells Bob her outcome, he can confirm correlations. She can even teleport a quantum state to Bob if she tells him the measurements she made and their outcomes, and he applies an appropriate untiary to his qubit. But this is only with an accompanying classical signal! But neither Alice nor Bob can learn anything about other qubits that may or may not be entangled to their own simply by measuring their own qubit! Even if Bob knows which Bell state he shares with Alice, he can never tell what she measured or even whether she measured by looking at his own qubit.
Even more importantly: The state of the universe evolves deterministically and locally! Thus, Einstein was correct that God does not roll dice, at last as far as the state of any closed system (including the universe) is concerned. And that there shouldn't be spooky action at a distance nor violations of locality. It turns out, that is all true of quantum mechanics in the MWI perspective described above! It is only our experience that seems random, but nature doesn't have any obligation to care about our experience! There is also no "collapse" in this picture, avoiding the well-known "measurement problem" entirely. And this resolves the "paradox" put forth by Einstein, Podolsky, and Rosen (EPR): Measurements of an entangled state cannot send any quantum information on their own nor violate locality in any meaningful sense!
