If you want to use your bike to get around with, then a geared bike with bigger wheels is better. It will be faster, less fatiguing and offer greater rider comfort. But if you want to do BMX style riding, with jumps and tricks and that, then a BMX is a better choice.
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Depends on how you're riding right now. If you're fast off the line and out of corners, but spinning out(not being able to pedal faster) on the straights, then go for a smaller rear sprocket. If you're gaining on the straights but losing ground coming out of corners go for a bigger sprocket.
Not really.
If it's meant to be a race bike, gears are not allowed. If it's meant to be a dirt or a vert it won't have them, partly because gear hubs are more easily broken and those bikes will get thrown around a lot, parrtly because there'll be some tricks that can't be done on a geared bike.
It is possible to buy geared bikes that look like BMXes, but odds are they won't hold up well to proper BMX use.
It's a bit of a strange combination too. BMXes aren't built to be ridden long distances in varying terrain(you can't sit and pedal very well). And geared bikes are meant to be good at exactly that thing - longish rides in varying terrain.
??? A BMX is Single-speed, it only has one gear, and that's what it's on unless it has lost the chain.