DIY BGA soldering feasability
I've heard of people doing it successfully, so it can be done, with some care and the right equipment. When the solder paste melts the chip aligns itself better than a fine-pitch QFP, so it can actually be easier than the latter. I'd be happier about doing it with an FPGA, as any missing connections could be corrected by rerouting the signals. The PCB has to be properly designed, of course.

I'd start with a QFN chip, they have some similarities to a BGA. If that works, I'd have some confidence that a BGA would be successful.