2 Questions - Could headers cause starting problem/ Running truck with no Thermostat

[eastcoastcivil]

I have a 67 C20 with a 350 4 bolt in it and Hedman headers. When I drive for any period of time where the truck will heat up and then turn it off I have to wait till it cools down a bit to start it. I would love to be able to turn the key and go but when I do there is no response...not even a click. Could my headers be heating up the solenoid and causing it not to start? Maybe I need to fab a heat shield! Anyone ever had this problem?

My truck was overheating a bit so I decided to replace the thermostat. I bought one at Crappy Tire and installed it last saturday...it didn't seem to be working since the truck was still running kind of hot so I took it out completely. I went for a drive on the highway and then into the city and it seemed to run around 180 degrees. Is it a good idea to run the truck without the thermostat? Any pros or cons? Things I should watch for?