Yes, if the fuel pressure regulator is failing to maintain proper fuel pressure, the engine will cut off. A weak fuel also do this. I recommend doing the proper tests below to verify whether the fuel pressure regulator or fuel pump is at fault.
The bottom portion of your question just showed up on my side. If the low oil pressure light is coming on, and the enigine is not turning over freely at times, you need to have the oil pressure tested. You may have a failing oil pump, or oil sludge build-up in the engine causing this problem. This needs to be lokked at as soon as possible. Furhter driving may cause the engine to lock up