a) When we place a limit order in the pre-market session if the order is not executed, will this order be valid for regular market.
b) Which limit order get a time priority in execution in the regular market (9.15 am to 3.30 am) , After market Limit order or the pre market limit order which is carry forwarded to regular market session.
Vinay,
a. Yes all pre-market orders will be valid for the regular market when it opens at 9.15 am.
b. This is a very interesting question, 
, The pre-market order would get priority over the limit order placed after market opening.
Since the broker submits the pre-market order between 9:00 am to 9:07 am to NSE, the timestamp of pre market orders will be earlier.
The broker submits the After Market Orders only from 9:15 onwards, so the timestamp in after market orders will be late compared to pre-market orders.
Usually NSE communicates the order confirmation of premarket orders, during 9:08 am and 9:15 am, so your order is already with nse but waitlisted tor regular trading which opens at 9:15 am.
If your broker software shows the pre-market order very late, it could be problem with your broker only.
Obviously pre-market orders will remain valid in regular market if it opens at 9:15. As we know order of aftermarket is normally submitted on 9:15 that is late in comparison of pre-market that’s why aftermarket order’s timestamp will also be late.
With another Broker house, i found the pre-market order getting listed in the order book very late. By that time, I had opened and closed a position in my trade… Probably their servers are slow. I stopped placing pre-market orders after that day, since i ended up with a open position which wasn’t needed so late after the volatility had ended.