Yes. FR code boxes are shorter geared as memory serves though. Yours will have a JQN or KDU code box.
There are 4 gearboxes listed for the BWJ engine.
JQN .... 11/06 - 06/07
KDU .... 06/07 - 02/08
KPA .... 03/08 - 05/09
KZW .... 05/09 - 05/11
The BHZ code is an engine code and that code belongs to an S3. If you plan to fit an S3 gearbox, then you should perhaps consider some other things like a diff and some driveshafts
You need to find a gearbox from a Cupra or Edition 30 ideally for gearing although an FR/GTi box will fit the engine - not sure if there is any other difference other than the gearing though - ie strenghtening. Also, I believe the gearing in the facelift Cupra is the same as the pre facelift FR but not 100% sure.
No - you need to find one as per the chaps list above or per my link on the post I did about part numbers. Your best chance to find a proper box is through a find a part thing, or call the likes of Stattlers who keep those things in stock