Can quantum systems interact with multiple environments of different types? If it can, how can we write the Hamiltonian of the total System is it just (for example with N bath)
$$
H_{tot} = H_{s} + H_{B_{1}} + H_{B_{2}} + ... + H_{B_{N}} + H_{I_{1}} + H_{I_{2}} + ... +H_{I_{N}}
$$
Yes, a quantum system can interact with multiple environments at the same time, and, if so, it will be described by a Hamiltonian such as the one you have written. Intuitively, a single quantum system may be in contact with different environments, such as a phonon bath, the electromagnetic fields, thermal collisions with external particles, etc., and each of them would be described by an infinite number of modes, coupled to the system with spectral densities of different shapes. We may even couple a single quantum system to multiple thermal reservoirs and the Hamiltonian formulation would be equivalent.
And can environments be a different structure, for example a qubit modeled by two level system coupled to a bosonic bath (harmonic) and to a different two level system (spin) independent from the bosonic bath ? How it evolves (finding the Linbladian operator) ?
Yes, in principle you can do whatever you want with quantum systems and reservoirs; you can connect them through different couplings and play around with different interaction-network topologies. The example you are proposing may be described by the Hamiltonian:
$$
H=H_{q1}+H_{q2}+H_B+H_{q1q2}+H_{q1B},
$$
where $H_{qj}$ is the free Hamiltonian of qubit $j$, $H_B$ is the bath Hamiltonian, $H_{q1q2}$ introduces a qubit-qubit coupling and $H_{q1B}$ is the interaction Hamiltonian between qubit 1 and the bath. Note, however, that in such systems the action of the bath may "reach" qubit 2 as well, if they qubit-qubit interaction is not very weak. In these cases, so-called "global master equations" are required. On the contrary, if $H_{q1q2} $ has a very weak coupling energy, you may rely on a local master equation and the dissipative action of the bath will only affect qubit 1, while qubit 2 will follow a unitary dynamics. This kind of systems are very interesting and may display curious phenomena, for instance, if you are interested in a reference, have a look at this paper.
Finally, this is not only about funny toy models. Here are a couple of references on experiments with multiple environments acting on superconducting qubits: https://doi.org/10.1038/s41567-018-0199-4 and https://doi.org/10.1038/s42005-020-0307-5.
