92 Integra Start problem

I have a problem starting my integra., it is automatic. It seems to only happen in the cold weather. I will turn the key and everything lights up but when I try to start it I get nothing. No noise, just a click sound. I will turn the key about two or three times with the same result, but then it will start like normal. Another problem is when I put the car from park to drive it is fine, but when I go from drive to reverse, it takes a couple seconds to kick in. When I go from drive or reverse to park, the same happens, it takes a up to a couple minutes to kick in and I can't get the key out because the car doesn't read that its in park. Any suggestions?

get a new shifting switch or whatever its called. but its down by the shifter just take it apart and there should be a switch that moves with the shifter. or it could possibly be the trans computer