Van will crank but will immediately sputter and die?

When it's cold (35 degrees or lower) the van will crank but will immediately sputter and die. The only way to keep the engine on is to give it gas. You have to give it gas for a while before it will stay on by itself. When you put it in reverse it will die. If it doesn't die then, when you throw it back into drive it will die. If it doesn't die then, and you drive it a couple blocks and you get to a stop sign, it will die. If it doesn't die then you might be ok.

This morning I started the engine, gave it some gas until it stayed on by itself. I let it idle in the driveway for about 10 minutes to warm up. When I came back out to load up, it died as we were getting in the van. When I tried to start it again, the battery was completely dead. No clicking sound. So I jumped the van and gave it gas to get it going again.

Side notes:
1.) I have noticed when you are driving, and you slow down to idle or stop, the air coming out of the vent will diminish and the volume on the tv that's in the van will lower. When you step on the gas again, they both pick back again where they were.

2.)Sometimes, after you finally get the van happy enough to drive, you will be sitting at a stop sign or light and suddenly you will feel it lurch forward and the engine will pick up in sound and you will have no problems whatsoever with it for the rest of the trip.

3.) It is leaking transmission fluid. Just drops, not flowing.

What in the world is wrong with my van?
Thanks!