Something electrical is wrong.

So, I was planning (still am) to help a friend of mine fix up his 95 Miata (a/c, no abs). I was thinking it would be mostly cosmetic stuff, but then the alternator went bad, so I replaced that and now everything is wonky.

Existing condition, driver side headlight doesn't go up and down correctly, the positioning is always off when you can get it connected to the arm that rotates it.

Basically, I replaced the alternator, got it all put back together and everything seemed fine, started it up, sounded a bit loud. I put the clutch in and everything quiets down a bit. Then I put it in gear and right as I let the clutch out, the headlights turn off, the dash backlight gets brighter, and the windshield wiper turns on (in off position).
Also, the headlight motor on the driver side was incredibly hot when the car was running and smelled like it was cooking itself.

I'm planning to check all the grounds, but I just wanted to see if anyone has any suggestions or has dealt with this before.

I'd check the fuse boxes and look for any torn up wires on the harness that might be causing a short or grounding out.

I'd say you're on the right track to suspect the electrical grounds. An expert once told me that weired electrical problems are frequently a grounding issue. A quick fix could be to run a jumper wire from the engine block to the frame and see if it makes any difference.

turns out the new alternator had a bad regulator and the old battery was dead. Replaced alternator again, and put in a new battery, and now everything works fine. Except for the radio (no longer turns on), the cigarette lighter/12v nolonger gets power, and the tachometer likes to delay under acceleration then drop down to indicating idle as soon as you aren't giving it gas. As soon as I floor the gas again the tach hesitates then jumps up to around where it should be...

I don't know about '95, but if it is the radio code, check this out:
http://www.miata.net/faq/faq3.html#L

thanks, should have mentioned it's an aftermarket unit. I need to pull it out and check if it has an inline or internal fuse added, all the fuses under the dash on driver side are good.