Sounds like the mode door or actuator is busted for issue #1. Don't recall how involved of a process it was to pull the dash on a gen 1 (I pulled my mom's 94 to paint it, but it was a while ago) or if you can even access the actuator when you get it off. On my gen 3 the tip of the door where the actuator attaches sheared off on my temp door, so no matter what I set it to I got hot air all the time. Sounds like a very similar problem, though it could be the actuator just isn't moving at all on yours. Can't really tell without taking it apart to watch it.
Not sure on number 2. Sounds like maybe a faulty sensor? Like it won't arm the alarm because one sensor still shows as "open" to the computer. Do your curtesy lights work when either door opens? I think there is a hood switch as well if I remember correctly. You might check those with a multimeter and see if they are opening and closing correctly. I think there was a test procedure you could do with some variety of key switch on/off thing and the fob that would let you test the switches one at a time. Something like turning the key on and off a few times would put it into test mode, then every time you open a door or switch the horn is supposed to honk once or something like that.