Ok, first thing i would do is get a multi meter and connect it inline to get a current draw reading. (Without a doubt though, if the rear heated screen light is constantly illuminated then the rear heated screen is ON), and firstly unplug the rear heated screen relay, you will probably find that the reading drops very low then, you will always have a small reading due to the immobiliser.

The rear heated screen fault could be a number of things, at a guess the relay had welded its contacts together, or there is also a possibility one of the door looms has developed a fault inside and you are getting a backfeed through that (heated mirrors being controlled by the heated rear window).

As for the alternator fault, ideally you want to try and at least kill the other faults (if you are not able to fix them) so you can at least check all the basic charging circuits. Check the thin blue wire on the alternator for dry/poor/broken connections, and also inspect the main charging cable that runs to the starter, its quite common for it to break down over the years, the fact the bulb is glowing shows a very slight difference in potential between the ignition and charging circuits.