The oil/grease has dried up, you've gotten water/dirt/rust inside, it's lost adjustment and/or wore out. Really no way of telling w/ taking it apart.
I'll post a link on bike maintenance for you.
If you have a flip-flop BMX hub, you can put a 14T on the smaller diameter side of the hub. If you have a cassette hub, you can get as small as 11T.
Depends on the bike frame..
change out the hub
There are two different hub designs for BMX, freewheels and freehubs. You need to know what you have if you want to replace the sprocket by the rear wheel.
No. a 9T sprocket will only fit a cassette hub, and not freewheel hubs.
wheel hub
Yes you can.... you can put any hub, spoke, and rim combinations together...maybe a few ecceptions but you can deffinetly do that... just bring the parts to your local shop and have them lace it.
If you have the right kind of rear hub and good balancing skills - yes.
replace hub or wheel bearings.
Kinda, sorta. You can certainly (have someone) build a BMX wheel around a fixie hub. I suppose you could somehow disable the freewheeling action of a regular hub too, but I wouldn't recommend it. Putting back pressure on the pedals on a bike that hasn't a real fixie hub can cause the sprocket to unscrew.
Probably. Today BMXes can have either a freewheel hub or a cassette hub, but as long as you stay within the hub type you can move parts from one bike to another.
You need the right kind of hub for that. If you haven't, buying a new rear wheel - with the right kind of hub - is probably the sensible option.