Q: Car won't start after seeing engine hot turn a/c off warning light.

I was driving my car on the highway when a warning came up on the dash saying engine hot a/c off. The car was still runnining fine and was able to make it home. Once i parked the car and turned it off the fans stayed on. I started the car again which it did start but i turned the car back off. I waited about 10 mins came back out to start again and the car wouldnt start. When i turned the key i was only getting a clicking sound but the lights and radio still came on. The next day we tried again and this time the car wouldnt come on at all. 2 days ago a friend came over to see if we could jump the car to see if it would start. He unhooked the cables to my battery and attached them to a spare battery. The car lights and radio came back on but would not start. But when we put the cables back on my battery the car wouldnt turn on at all and then the key was stuck in the ignition. We had to hook the battery cables back up to the spare battery to get the key out. Could it be the battery?

My car has 109000 miles.
My car has an automatic transmission.