How do you program the PIC? How do you know that you are actually programming the PIC?
Does it read back with the same code you programmed it with?
Exactly. I worked fine so far therefor there are no issues here.

Have you even considered changing the program to the one I suggested? Or are you still running the same 'one-shot' 'turn the leds on once then turn them off' program?
Yes, I have already considered your program and I decided to stick to my program.

Do you have a decent cap across Vdd and Vss on the PIC itself?
I just connected a 100nF capacitor in between Vdd and Vss. Nothing.

Is the power supply stable enough? Does it rise to +5v smoothly? Does Vpp come up to +5v before Vdd?
I am using a 13.8 volts dc regulated power supply aswell as 3 times 7805. One 7805 for the pic itself and two parallel for other items like leds.

Do you have a 'solderless breadboard' to pull the PIC out of that circuit board and try it elsewhere?
Yes. I just did. It is working there.

Is the PIC installed in the correct orientation?
Yes.