Re:
a wobble in the back wheel shouldn't affect your gear changing.
have you checked your chainline? does the new bottom bracket have the same length axle as the old one?
personally if you're replacing the rest of the drivetrain then I'd replace the cassette too, a new chain on a worn cassette can make the chain slip or jump about.
I use a chain wear indicator now, cost a fiver and tells me when it's time to change chains, rather than running them into the ground and having to replace the cassette at the same time.