bad battery

hi, battery was dead yesterday morning so i jumped it and it started right up. drove to work. started it 4 hours later and it started right up. i came out about 5 hours after that and dead again. so jumped it and drove home. so by this time i am thinking bad battery. this morning i had to jump it and drive to work. let it sit for 6 hours and then i went out and tried to start it and it fired right up. but i'm still thinking i have to buy a new battery, the real trouble begins i left work went to pick up the kid and i shut it off for about 10 minutes. i started it and pulled away and at the same time i clicked on the air conditioner, as soon as i turned it on it started beeping and said engine overheat, so i turned it off for a few minutes, when i turned it back on everything was normal. about 10 minutes later i was sitting at a read light and it started to overheat again but didnt quite hit the red, the light turned green so i went around the corner and pulled into the parking lot and everything went back to normal. i guess my question would be is it the bad battery thats causing it to overheat. i thought maybe somthing wasnt turning on or not running right. the other funny thing i noticed about 2 weeks ago every now and then when i turn it off a few seconds later the fan will kick on and run for a few minutes. it never did that before. any help or advice would be hugely appriciated. thanks in advance