Many early cars had the gas tank right at the base of the windshield, above the engine. You can imagine the risk of fire if fuel was spilled on top of a running engine.

Cars can backfire at idle, possibly sending embers out the exhaust, a spill while filling a tank can put fuel on the ground, the fuel vapor that rapidly develops can be ignited by hot soot particles if the engine backfired.

These things are not much of an issue nowadays, but fuel spills still create a serious hazard and it's definitely safer to turn vehicles off for this reason.

For the last 40 years or so, vehicles have had sealed fuel systems for fuel vapor control. An emissions system. If the car is running while fueling, the system will detect a leak and a check engine, ses light will be illuminated.

There is no way that filling the tank will change the output pressure of the fuel pump. I'm sorry but nothing you described here is accurate.

Was a food court / hotdog stand kind of restaurant. Can't remember the uniform perfectly but her shirt looks like it.

Did she work at hotdog on a stick?