Q: Car starts, but quits; removing/replacing battery cable helps

The check engine light came on around two weeks ago. I checked the usual things: gas cap, changed the oil, etc. Other than the warning light, my car was running very well. Then a few days ago, I hit a massive pothole which destroyed the tire and the alignment. We did get all of that repaired and fixed. Then later, the car began having noises like an RC car. I was stopped at a store and when I tried to backup, Reverse didn’t want to function, so I put it in Park and then back into Reverse and it backed up fine. Thirty minutes later I was driving home with no problems, but yesterday the noise returned and the car was not shifting properly. The car would start, but then quit. I removed the battery cable, replaced it, and the car drove back home in good performance. Then I decided I should see a mechanic, so I took it in last night and the warning light came back on. Then it would not shift and would not pick up any speed. The mechanic's shop was still 15 miles away. After sitting a while, I removed the battery cable and then reinserted it and I got to the mechanics without any more problems. The mechanic told me “complete fail code” but nothing else. He still has the car. I'm thinking that it might be a sensor/computer problem and from what I've learned, if it were the transmission then it would always shift and always lose speed, even when removing and replacing the battery cable. Would you please give me your assessment of my car’s condition and what will repair it? Thank you.