1. Does the fuel pump hold the pressure in the fuel lines when the truck is not running?
It should
2. When a fuel pump goes bad, can it be gradual thing or does it just stop working?
yes
3. Does the fuel pressure regulator hold the pressure in the fuel rail/lines when the truck is not running?
It should if the pump is not back leaking and the injectors are not leaking
4. When the truck is not running, can fuel injectors start leaking and therefore causing the loss of fuel pressure?
yes

The fuel system is a loop, with a pressurized section from the pump to the pressure regulator. The filter is in the pressurized part. Most pumps have some kind of check valve. The regulator should hold the spec pressure. The injectors should not drip at the rated pressure.

In a pinch, I've used a hand vac pump to test injectors (on the pintle) and FPR's on the outlet pipe. I've successfully culled thru JY injectors that way. The only definitive test is to put a pressure gague adapter on it at the fuel rail inlet and watch the gauge on shutdown.