Radio and Interior Lights Never Turn Off and other Gremlins

I just got a very low millage 1992 LeBaron convertible with the 3.0L. While
looking good and running perfectly I have a lot of electrical gremlins I hope
you all can help me with. Some or all issues

1. Headlight Doors don't open or close, just swing freely. I hear what sound
like a motor run for a little when I turn on the headlights. (I am guessing
some linkage disconnected, mechanical Gremlin)

2. When I turn on the headlights they come right on as well as the dash
lights and the headlight indicator comes on the dash. When I turn of the
lights the dash and indicator go off but the headlights stay on. I then have to
restart the car with the light switch in the off position to turn off headlights.

3. The Speedometer doesn't read anything, all other dash items work
perfectly.

4. The radio and all interior light (except dash)  don't work at any time. I
found the fuse was missing so I replaced it. Now all of the interior lights and
radio work great however never turn off so I pulled the fuse back out. This is
with the key in or out didn't matter. (I was thinking failed relay in the on
position but don't know of a relay that controls that combination of items)

Have not found any other issues yet. Thanks for your time and brain power
to help me on this project.