What you're talking about is called a 1x9 setup. A 9 speed cassette will work in place of an 8 speed. All you'll need is the narrower chain for 9 speed and a new rear shifter. For the front, ditch the granny gear, replace the big ring with a bash guard/rock ring. You might also want to ad an n-gear jump stop chain guide to make sure the chain can't jump off to the inside (the bash guard keeps it from jumping off to the outside).

No, you can't use a bmx chain with a 9 speed cassette. It is too wide.

Just curious, why change to 9 speed? 8 speed can cover the same range and is more robust, since the chain is wider... I usually run a 1x8 (or even 1x7) for my projects. Have fun.

-Joe