2000 GMC Safari strange set of problems

I have a 2000 GMC Safari with 78k miles. It has a bizzarre set of problems, but the most troublesome one at the moment is that the battery keeps dying. Sometimes it starts, sometimes it just clicks. Sometimes it won't start after I've been out running errands and I'll run in and out of a store and poof - won't start. It happens more frequently when it is cold. I have a spare key and just leave the thing running if I'm not going to be long. I keep all the interior lights off because, for example, if I leave the dome light on when I'm taking groceries out of the car or something like that, it won't start next time I try. Also, the fuel gauge doesn't work (jumps all over the place), the driver's side window only goes up fractions of an inch at a time, the AC died and there is a continuous flow of really really hot air blowing on the floor of the passenger side (regardless of what I do with the controls). Do I have a lemon? And any suggestions on the starting problems? I've had it in the shop many times and nobody seems to be able to find anything. Help!

Sounds like a battery or battery connection problem. I'd start there, have it checked.

With the exception of the heater fan problem, we had very similar issues on our 2000 Safari and it turned out to be the alternator. It eventually died.

Sounds like the alternator.

I have a 2000 GMC Safari with 78k miles. It has a bizzarre set of problems, but the most troublesome one at the moment is that the battery keeps dying. Sometimes it starts, sometimes it just clicks. Sometimes it won't start after I've been out running errands and I'll run in and out of a store and poof - won't start. It happens more frequently when it is cold. I have a spare key and just leave the thing running if I'm not going to be long. I keep all the interior lights off because, for example, if I leave the dome light on when I'm taking groceries out of the car or something like that, it won't start next time I try. Also, the fuel gauge doesn't work (jumps all over the place), the driver's side window only goes up fractions of an inch at a time, the AC died and there is a continuous flow of really really hot air blowing on the floor of the passenger side (regardless of what I do with the controls). Do I have a lemon? And any suggestions on the starting problems? I've had it in the shop many times and nobody seems to be able to find anything. Help!
if you were to read a number of the other postings in this forum you would read a answer to a number of your problems,i just changed my blower motor,thermostat,and water pump,my van runs great now that it is not running as hot.just read the forums.