Some other people have been dancing around this but I am going to come right out and say it: I think you have bitten off more than you can chew and you should probably not be attempting this project as your first PCB.
You should really know why 13 layers is not possible. Ever. (Yes, some vendors will quote it, blah blah blah, some might even do it... but the quotes are f-off quotes and the boards will fail early if anyone is dumb enough to build them for any reason short of total desperation.) To get 134 I/Os out of a 676-ball BGA is not very difficult. You can probably do this in 4 layers if you are clever and patient; 6 is probably not too hard at all; and 8 should yield an excellent electrical and physical design, executed quickly, for the modest unit cost increase. (So, what I'm saying is, start with 8.) The only thing that should give you real pause is the fanout, as you do mention... but it's an FPGA so you can just put everything on the outer two rows (there's ~200 balls there!), where
no vias are needed to break out, or at least you can break it all out however it needs to go to be nice and clean.
Your design rules may even permit vias directly between pads, at 1mm pitch. (1mm pitch is easy mode.) Or you can probably get three traces between pads. This is what makes these things possible.
I think there is some ESD protection within the FPGA.
There is not. Or, at least, there is little enough that you will be happier pretending there is not. Protect all off-board signals if you want your chips to live!
My concern with a single GND pin is not related to DC current capacity, its to do with signal integrity.
So here I have a 14-bit 120MHz DAC, AD9744. It only seems to have one or at most two ground legs. Why it doesn't need a ground leg for each signal line like the guy from stackexchange suggests that you posted? I'm a bit falling behind on theory here. It's obviously not a huge PCB, only a small silicon chip, but still
They can get away with this for ICs because ICs are tiny and sometimes even have onchip decoupling.
You (and I) cannot get away with this for connectors. Connectors are hell, electrically, mechanically, and every other way. Your precious signals need their figurative hands held every step of their journeys off board, or they will suffer, and your reputation will suffer for it.
You may want to post your schematic here (as a PDF! or PNG! or whatever that isn't proprietary to your software!) for comment (probably in another thread). Someone might be able to save you a spin or three.