It still sounds like an LVP problem to me. When LVP is selected, PORTB bit 3 becomes a selector pin for programming (=1) or normal mode (=0). If it is left floating, or the internal PORTB pull-ups are turned on, the PIC will not run and you risk corrupting the program. To see if I'm right, connect a wire between PORTB.3 (pin 36) and VSS. If that makes it work, you can leave it like that or turn the LVP mode off. Note that you can't use PORTB.3 for anything else while you are in LVP mode, it becomes dedicated to selecting programming mode.
Your schematic is also missing two vital components. You should have two capacitors (suggest 100nF) wired directly between the VSS and VDD pins, one on each side of the PIC. They should have wires as short as possible.
I can't see the green electrolytic capacitor on the schematic. Where is it wired?
Brian.