3. https://pulumi.io/reference/cd.html#branching-strategy-for-deployments suggests having git branch per stack and merging between branches to promote changes across environments (e.g. dev -> staging -> prod). I guess it would also be very tricky in my case but I would like to confirm how this is done in deployments like in my scenario.

That doc just calls out one approach, we've definitely seen many others in use. Really depends what you want to do exactly - if you have a git repo per program, and only one instance of that program (one stack), then you can just map

master

of each of those repos to that stack.