It's all is possible. But: check the synchronization guide
at first. It's not that simple as you may imagine.
So answering your questions in detail:
Use my own registration page and copy user records over at signup
Copying is a sign of a really bad lame programming. Your system should use only one members table. miniBB then could use such table as described in the Guide above.
Use my own login page and automatically log the user into the forum
It more depends on your own script, how it sets cookies. miniBB could then read such cookie and verify the user based on your members table, and then log user in or not. In general, this based is the same practice as used above. So it's possible.
Remove Login/Logout/Signup menus
Removing is the most easy part, just modify miniBB templates. To remove the Sign up, set