Should I add GND to this?
It won't work if you don't.
Is it safe to add RESET line to the header and connect it to RESET on Arduino board?
Yes providing the grounds are connected together and they are both running off the same size voltage (they don't have to be the same supply but they need to be within 0.5v of each other)
Should I set any protection (diodes etc) on this lines
No need if the power supplies are the same.