PIC MCU PIC16F887 not working with external power
The power source has nothing to do with debug vs. release. You can set the debug tool to supply power or not. The debug release requires the debugger attached to the data and clock lines in order to run. Release should not.  

You should connect all the Vdd and Vss pins for reliable operation. You should have a resistor in series with the LED. In addition to input and output caps on the LM7805 regulator (not generally required for stability but recommended), make sure you have an appropriate ceramic bypass capacitor (eg. 100nF) near the power pins of the MCU. 

Finally, and most likely your proximate problem, if memory serves, turn off LVP (or add a 1K from PGM to ground).