My a/c quit blowing cold one day. I switched the a/c on and off and got it to work again. This went on a week. Then my headlights and a/c started staying on without the keys in ignition and the compressor would run without keys in. The battery died because of these problems. Any help?

Welcome to the forums.
I'm moving your post to the proper section, I assume you are asking about the 2004 Outlander.

1. Light stay on:
Please check your Owners Manual for "Follow me home" or something-like-that headlights function. I am not sure this is what is causing your problem, but may be you find out, that your car is just doing it's job. I think different sequences of shutting off the ignition, opening and closing doors and switching lights are enabling some hedalight features.

2. Well, the AC compressor is not electrical one. It is running only when engine is running. You're probably talking about the fans in front of the car - cooling radiator fans. There is a known problem with temp sensors relaying power to those fans. There was a recall for this fault. Can you doublecheck what is draining your battery and in case you confirm this problem, can you visit mitsu dealer?

Rgds