Heck, a fuse could be bad, but still test good, for that matter.
How do you know that the fuel stops?
Things to check? Well, starts and dies could be a clogged exhaust. No fuel shouldn't cause the alternator light to come on. I'd check the alternator.
I'd check the ICM, if this still has it. You can check fuel pressure, and the injectors.
Hmmm. Temp rising. Still thinking exhaust blockage. I know it's been replaced, but other than running lean, the temp shouldn't suddenly rise, unless the coolant isn't flowing. If the coolant stops, that's the pump/belt, or a leak in the system. You could check the timing, but I suspect it'll run. That would cause starting issues and temp issues.
By my understanding, for this to occur and the relay be an issue, the oil pressure sender is also a problem, but you didn't report low oil pressure.
Was this the car that has the thingy to collect some fluid or other?