I just watched the video....holy smokes, is the engine even bolted down? It should not move that much when being started. Looks like it'd rotate all the way over if it actually had a load on it. Mine doesn't even wiggle, I don't use the lexus mounts though specifically to avoid the engine rocking like that.
Back to the problem at hand though, the little nipple behind the AFM that you mention is open...cap it. That's the one for the power steering idle up, and many people delete it anyway. The ECU can't properly cope when it doesn't know how much air is going through the engine, and you'd be surprised at the massive difference a vacuum leak that that can make.
After the engine dies, is the fuel pump still running? You mentioned a wiring issue to the fuel pump, how is it wired now?
If you're just using a fuel pump relay instead of the fuel pump computer, there are two pins to the fuel pump computer, one tells the pump to turn on, (FC I think, maybe FP?) and the other tells it how fast to run(FPR). I don't have the schematic in front of me right now though. If you have it hooked to the wrong one, the fuel pump will only work while the ECU is telling to it to run at high speed. (During startup and under heavy load.) So as soon as the ECU thinks it's slowing down the fuel pump, it's actually turning it off.
Mine just runs through my circuit open relay, but my 1UZ is in a toyota. Because the Lexus fuel pump computers seems very prone to failure, and it presented itself as one more headache I didn't want to deal with. So if you are using the Lexus fuel computer, maybe look to make sure it's working. God knows they fail almost as much as chevy trucks.
