Q: Car starts cold in mornings, but cuts off in a minute. Will start back up after a couple of hours but immediately cuts off.

I bought the 2002 Eclipse a couple of months ago. Shortly after it cut off while I was sitting idle at a store. It immediately started back up but cut off two or three more times on the way home. It was harder and harder to restart each time after the first time. I killed the battery trying to get it cranked and a battery charger was connected backward to the battery and the factory alarm started going off. I had it towed to a shop and they pulled a horn fuse and relay and said reset the computer. It sounds like the hazard lights are on but they aren't and it stops after a minute. The car started skipping some and then started the same initial problem as before which I've described. New fuel pump installed. The lot said the distributor cap was replaced right before purchase. New battery terminals and new positive cable fuse put on, too. Suggestions, please?

My car has 200000 miles.
My car has an automatic transmission.