Probably not. BMX bikes usually have freewheels while MTBs usually have freehubs, and these don't mix. You might be able to use parts from a single-speed MTB on a BMX though.
It will if you have bmx (or bmx style) cranks.
it depends on what make and model your BMX or MTB is
the easiest way is to go to your local bike shop
A racing bike, a mountain bike and a BMX bike.
Not a sprocket only, but you can buy a 23-tooth freewheel that should fit.
You might have to either lengthen or shorten the chain, depending on if you're getting a bigger or smaller sprocket.
A good bmx bike will range from 20-25 lbs, have a small sprocket (cog) either 23 teeth, 25 or 28. It will also look nice have no or one break.
Depends on what type of riding you want to do. For BMX-type stunts - get a BMX. For going some serious distances both on and off-road - get a MTB. Don't really know what you mean by jump bike.
Yes, if you can find the correct bottom bracket for the frame.
Two, but this is not the same as how many speeds the BMX bike in question has. A BMX bike has a sprocket in the front, and a cog in the back and each is a gear, but is considered a single speed bicycle because two gears and a chain are required for transmission of rotational force from the crankset to the rear wheel.
it would depend on whither you are riding on a mountain bike or something that wont be doing stunts or a bmx or dmr. for mountain bike it probably will not matter whither it is big or not however for bmx or dmr it would be a good idea to have a smaller one as when at skate parks eg. you where going to drop into a bowl that is quite steep if you have a big on then the chances are it is going to hit the coping (top of edge) and maybe fall off. also if you are thinking of getting a smaller sprocket you will need to get a smaller cassette but don't guess how big the cassette should be for a sprocket just take it to a bike shop and they should be able to tell you. for a smaller sprocket i would recommend a 25t to 30t,, the 25t will have to have about an 8 or 9 tooth cassette. hope all of this helps took me a while to type out anyway :P!!
There are MTBs with 20" and 24" wheels for smaller riders, which probably is the closest thing to what you're describing. These wouldn't be BMX bicycles though, since BMX bikes don't have gears and have a taller handlebar than a mountain bike.