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.
As part of my old job I had to build some boards with the Foxconn 478 pin BGA socket for the old P4s. It was the first component I put on the board. My technician and I did it by hand, using a little jig he made to line everything up right. You'll need a mask for screen printing the solder. I used post-it notes to set the snapoff distance, I think it was 2 or 3 post it notes thick. Reflow with a toaster oven (I had a nice reflow oven back then).
We yielded about 8/10 boards, maybe a little less. Usually the fails had everything off a row, but occasionally one would line up between pads and short everything (big mess, obviously my pad masking wasn't quite right).
So it's doable, by hand, if you are careful. Do it first, with nothing else on the board. If there are more than one BGA on the board your yield will be poor unless the pad spacing is large enough.
It just occurred to me that we could have tried buzzing out the board; there were a pair of voltage sense pins that we could have used. That might have prevented some of the off-by-one-row alignment errors and shorting that we experienced, but the sockets were only about 5$ a piece.