2002 Chevy Cavalier intermitant start problems

Electrical problem
2002 Chevy Cavalier 4 cyl Front Wheel Drive Automatic 160k miles

About five days ago daughter drove car to work shut off and went to lock door with power locks and would not work. She then put key in switch and nothing would come on, lights, dash lights, or radio. I went there and it started but surged for a second or two before smoothing out. Checked battery voltage while running around 13.8 checked at altenatoraround 14.1. When compressor was engaged idle would drop to point of almost stalling and engine would surge couple times and level out. Cleaned dirty ground at battery tray and symptoms stopped and was fine till yesterday. This time when I tried to crank no lights no power to interior and would not turn over. Left switch on and light switch on, walked away from vehicle to get cables to see if could jump start and lights came on and was able to start engine. Problem with compressor before no longer causing surges. Could problem be in battery? My guess is between battery and fuse panels. Thanks rjg71