A search on the radio issue here ( use radio battery in your search ) will throw up a number of posts on how to fix it.
Heres one that refers to the problem in this part of the website
In a nutshell, you need to rewire some part of the connector to the radio with a feed that is only on with the ignition - I think the usual one involves tapping into the feed for the cigarette lighter, putting a fuse on it and connecting it to the right wire that provides power in the Alfa plug.
Try searching in the ICE section of the forum - i think i found a lot of good stuff there. If I remember correctly the standard radio in the car would allow you to listen to the radio if the ignition was off but it would turn itself off automatically after a while - the replacement radios generally don't do this and drain the battery. I will have to do this myself shortly as I have no radio in my car that I bought during the week.
I also think the door issue will turn up as well, my drivers door looks like it will do the same as your passenger door, a search on Bowden cable door may show up your problem or lead you to the solution to your problem.
Last edited by NedFlanders; 18-11-12 at 23:55.
Reason: Checked that my search terms would work.