This all started a week ago. My battery seemed a little slow in the evening, then the next morning it was flat dead. I bought a new battery, installed it and everything was good. I parked the car after driving in traffic and the temp gauge was a little over normal, which it does some times, then I parked the car and it sat for 20 minutes. When I returned to the car, it turns over but does not fire. I let it sit for one hour and it cranked and ran fine. No other problems for a week. I drove the car in more traffic, the temp guage again was above normal, but no where near red, the car sat parked for
15 minutes and when I returned it wouldn't start for one hour. Last night, I had the same issue, so I let it sit for an hour and it cranked and ran, except this time I spoke with some people for about 10 minutes and the temp gauge went back up and the car turned off. Later it stopped running in traffic when the gauge registered a little over normal.

So, in summary, if it gets a little over normal in temperature, it won't start if it sits for 15-30 minutes. It will start after it cools down an hour or so.
The fan has never come on in the 4 years I have owned the car, A/C or not.
2001 E320 Wagon, auto trans, USA car