Any 20inch forks will fit.
That's a BMX, and there's not a lot of difference between forks for those. AS long as you get the correct steerer tube length, any BMx fork would give you a rideable bike.
Depends on what's already on there, what's available, and how much clearance the bike has at the fork and frame.
The main parts are: frame fork wheels (handle)bars saddle seat post cranks pedals
BMX bikes all have the same size (20") wheels, so they're very similar in size. Doesn't matter that much as they are mainly ridden standing up anyhow.
It depends on the length of the chainstays on your frame.
All "true" BMX bikes are built for 20" wheels, which makes them very similar in size.
No. A BMX is almost entirely ridden "out of the saddle" (=standing up), so size doesn't mean that much. (most) MTBs are intended for some seated riding, which makes frame size much more important.
It's 1/8" width, but the length is determined by the size of the bike frame, and the size of the sprocket/chainwheel you're using.
There's a small chance/risk that the wider tire might not clear the frame/fork, but probably it will be OK.
A BMX stem is what connects the handlebars and forks to the frame.
The headset is a set of bearings that allows the fork to turn smoothly in the head tube of the frame w/o sticking wobbling around when you turn the handle bars.
In BMX bikes there isn't much to choose from, they're almost all use 20" wheels, which means that there isn't much difference in frame size either. Since they're mostly ridden out of the saddle it isn't much of a problem either.