Should be moved to the Q/A thread... but I'm not an admin so here is an answer:
As an owner you can fire a coach at any time in the season or postseason and it will prompt a hiring process, if you fire a coach on Black Monday (first week of postseason) then you will be competing against other teams for hiring from the coaching pool.
If your buddy quits as a user created coach it should default back to having Mike McCarthy as the coach again, so yes you can take over as owner and he should be there. If not, as long as that coach isn't coaching elsewhere you can fire current coach and hire him mid season from the owner staff screen.
I turn coach firing off generally so I control the process somewhat, and then just demand release on a coach I want firing in the postseason and let the CPU owner make a new hire (as I am offline and don't want to lock out an owner).