Well. I'll try to help you with a couple things.
First as far as the heat goes. Does the blower not work? Meaning when you say you don't have heat, do you mean air is coming out but it's not hot, or is there no air coming out of the vents at all? If it is the latter and there is no air coming out, it is probably your Final Stage Resistor. Costs about 50 bucks online and is pretty easy to replace. Just search the site for a DYI on that.
Second. The cheapest way to test your alarm issue would be to just replace the battery in the key. It's not exactly the simplest DYI due to the fact that the battery is not removable and the key is sealed, but it can be done. You need to slice the key open using a blade of some sort (be careful), once you open up the key you will see how the battery is attached to the small circuit board.
You can then clip the terminal that is attached to the battery so you can remove it. (you might be able to use a soldering iron and heat it up and remove it that way, I forget honestly since its been about 6 years since I had to do this on mine. Once you have the battery out you can look up the type of battery and order a new one online.
I ordered mine from Mouser Electronics for like a buck fifty plus shipping. Then just pop the battery in and solder the terminals back to it like the old one and then be careful when closing the key back up making sure not to break any of the components on the inside. Reseal using superglue.
You COULD also just order a new key from the dealership which will run you about 250 bucks. But if you have the patience it is WAYYYYYYY cheaper going the DYI route.
Something seems fishy though to be honest. I can't imagine the car isn't supposed to disable the alarm when opening it from the key cylinder on the door. Who knows though.
Edit: Read another thread by you on the heat issue, It seems you are getting air. I know this may sound stupid, but there is a center control right between the 2 middle airvents that goes up for heat and down for cold air, sometimes people miss that thinking the electronic control is the only way to control the temp of the air coming out. Just figured I would mention that in case.
Last edited by Surrept; 02-18-2013 at 05:33 AM.