Most BMX have sprockets with freewheels as one unit. This means that you need a special puller to grip notches on the freewheel to be able to unscrew the old one. The new one can be started by hand and will tighten down by itself as you pedal. If you have a low-end one with a built-in coaster brake other rules apply and you're probably looking at something with a lockring arrangement.
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They are designed for 2 different riding types. If you have the ability the modify a steel, steel composite, or aluminum frame you can do it. But there is a urban sport type of bike out there called a 29er bike.
You can use a MTB any way you want. but being bigger it will also be heavier, more cumbersome and more likely to break if it gets tossed around. In short track racing, tricks and jumps it'll be much harder to handle than a BMX. OTOH it'll go faster and be better for distance riding.
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.
No simple answer to that. Both BMX and MTBs have several different types of bottom brackets. Some of these are interchangeable, some aren't.
Depends. There are several different sizes of steerer tubes, stems and bars. Some will work together, others won't. Depending on what parts you have, you may well succeed.