When you say add spin and angular momentum the following equation pops into my head:
$$
\hat{\boldsymbol{J}} = \hat{\boldsymbol{L}}+\hat{\boldsymbol{S}}
$$
which is shorthand for the three equations:
$$
\begin{bmatrix}
\hat{J}_x\\ \hat{J}_y \\ \hat{J}_z
\end{bmatrix}
=
\begin{bmatrix}
\hat{L}_x\\ \hat{L}_y \\ \hat{L}_z
\end{bmatrix}
+
\begin{bmatrix}
\hat{S}_x\\ \hat{S}_y \\ \hat{S}_z
\end{bmatrix}
$$
So the question "why can spin and angular momentum add" boils down to the question of why can we write down
$$
\hat{J}_x = \hat{L}_x + \hat{S}_x
$$
The short (kind of tongue in cheek) answer is that we can add $\hat{L}_x$ to $\hat{S}_x$ and define a new operator $\hat{J}_x$ as their sum because both $\hat{L}_x$ and $\hat{S}_x$ are operators which act on the same Hilbert space. Namely the Hilbert space which is the tensor product of the spin Hilbert space and the particle position Hilbert space.
$\mathcal{H}_{\text{tot}} = \mathcal{H}_{\text{mech}}\otimes\mathcal{H}_{\text{spin}}$
If we define $\hat{L}_x^{\text{mech}}$ to be the operator which explicitly operates only on $\mathcal{H}_\text{mech}$ and likewise for $\hat{S}_x^{\text{spin}}$ and $\mathcal{H}_{\text{spin}}$ we can write down
\begin{align}
\hat{L}_x &= \hat{L}_x^{\text{mech}}\otimes \hat{I}^{\text{spin}}\\
\hat{S}_x &= \hat{I}^{\text{mech}}\otimes\hat{S}_x^{\text{spin}}
\end{align}
These operators both act on $\mathcal{H}_{\text{tot}}$ so they can be added together.
There is a follow up questions is "why would we want to define a new operator which is the sum of mechanical angular momentum and spin?" This is a more physically (rather than mathematically) motivated question. The answer is that physically spin and angular momentum behave similarly. For example, they satisfy similar symmetry properties and charged particles with either spin or mechanical angular momentum both have magnetic moments. In the end the notion that spin is a type of angular momentum ends up being a physical intuition which is very useful. It then becomes useful to group all types of angular momentum into a single operator $\hat{\boldsymbol{J}}$ in certain cases.
One more note: I do not understand your notation $Y_l^m x(-)$ I don't know what the $x$ or the $(-)$ are refering to. Normally when I think of the spherical harmonics, $Y_l^m$ I think of a function of a polar and azimuthal angle, $Y_l^m(\theta, \phi)$ which tells you something about the real space spatial distribution of a wavefunction. In particular spherical harmonics come up when describing states with non-zero mechanical angular momentum. However, spherical harmonics are not necessary to describe intrinsic spin angular momentum*.
*Of course often spherical harmonics may show up in the discussion of spin angular momentum by way of noting that both spin operators/states transform in similar ways under rotations as the the spherical harmonics do.
