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The door lock issue is most likely caused by a bad relay in the door control module. If you are electricaly savy, you can disassemble the module and replace the relays. There is a post on here that explains how to do this and is VERY informative.
Here is a picture of where the module is inside the door.
My 02 Z06 LDCM
Here is a picture of the board inside the box. It was taken off the post i was refering to:
As for the extermal temp read out, you most likely have a bad temp probe. Crawl under the front of the car and look up inside the radiator shroud opening. On the passengers side you will see a small black pencil shaped probe sticking through the plastic shroud. That is the EXT Temp sensor. I have a picture of the probe somewhere.