Electrical system: Which is best, Lightly Toasted or Fried?

I think I lightly toasted mine.

Last weekend I was planning on working on my currently off the grid retirement home. So to charge up my power drill, I plugged in my 300 watt power inverter into the 3 port extension which was plugged into my lighter outlet and then plugged in the battery charger into it.

The red overload light lit up and smoke started curling from the inverter.  Quickly pulled the power plug from the extension, but things are acting strange now.

My radar detector, which was also connected to the 3 port extension now clicks continually. 
The rear wiper no longer works, but when I push the button to the first click, the gate ajar light shines and the clock display on the radio freaks out with gibberish and then goes black.
If I push it to the full on, the gate ajar light shines, clock display freaks out... and the low oil level light shines.

The clock will also reset itself, without the help of the wiper switch.
Everything else seems to be working fine.  Rear window release, power windows, locks, gages, cruise .
Do you think I toasted the computer?

First you need to find the common wires to the circuit. You may have a short. You may have also burned a ground off. When something smokes you have a high current and usually a short to ground. Using 2 power supplies for DC can cause the weaker one to draw from the more powerful one. The current heats other things besides the unit itself. If the wires get hot enough, the insulation melts, causing a short.

The clock, rear wiper, radio, gate ajar light, and radar detector are not controlled by the computer. The low oil level light is powered through the power train module in a 94-96.

I would start with the lighter, and work backwards through the electrical system.

My wife said, "Its the fuse!"  I told her, "If it was the fuse, we wouldn't have any clock, tailgate ajar etc. It's a short."  Long story short.  I braked for a stop light and someone pulls up behind me and tells me I have no brake lights, so since I was checking the in the fuse box anyway, I pull the #2 fuse and there was just a sliver of metal keeping the connection intact. I replaced the fuse, everything works, she is still gloating.