Entropy change during irreversible process
The answers to your first and third questions are &quot;yes.&quot;  Here is a tutorial on how to calculate the change in entropy of the system or the surroundings for an irreversible process:  https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/. The reference also provides worked examples.  Note that, in considering entropy changes for both the system and the surroundings from an irreversible process, one needs to first separate the system from the surroundings and then subject each of them separately to a reversible path.  These separate reversible paths do not have to match.
It simplifies things a little to think about how entropy of a system can change.  There are only two ways:  entropy transfer across the boundary of the system by heat flow in and out at the boundary temperature, and entropy generation (due to irreversibility) within the system.  In a reversible process, only the first mechanism is present.  In both cases, the transfer across the boundary is $\int{dq/T_B}$ where $T_B$ is the boundary temperature.
Regarding questions 2 and 4 (involving ideal constant temperature reservoirs), these are somewhat limiting cases.  Here, if you imagine a very thin conductive slab between the two reservoirs (with high temperature at one face of the slab and low temperature at the other face), then for the irreversible process involving the transfer of heat between the reservoirs, all the entropy generation would take place within the intervening slab, while the reservoirs would each separately experience only entropy transfer with the slab.  In the limit where the slab becomes infinitesimally thin, we would obtain the system you have envisioned, but all the entropy would still have been generated at the interface, and each reservoir would only experience entropy exchange with the interface.