Smoking from throttle body

About a week ago when I shut the car off I noticed smoke coming from the hood and a strong burning rubber smell. I immediately popped the hood and noticed it was coming from the throttle body (or very near it) and that my serpentine belt was extremely cracked and near snapping. We replaced the belt and hoped it would solve the issues (or at least the smell) while doing all this we realized we had no power steering fluid so we filled that up (admittedly over filled it and it spilled). After these few little checks we took it for a test drive, no where far, just around the block enough times to get it warmed up. The smell came back though it was a little less rubber like and more chemical like (we assumed it was just the over filled power steering fluid), but as soon as we stopped the car we noticed the smoke again. Same location.

A few other things I have noticed about this car that I will mention just in case it is related somehow. It does not like to shift to second on its own very often. We can usually get her to shift to third and fourth with no problems. (We assumed we needed a new transmission and we got one but have not put it in yet). Also, our indicator lights in the dash randomly come on and off with no apparent reason. Brake/ABS together. Theft/service together and check engine, oil, and Trac off individually. None of these combinations are ever mixed together and all disappear of we restart the car. We assumed this to be electrical as our stereo also resets itself when using blinkers sometimes.

Any idea what I should check? I hate replacing parts that are not needed just to test if it is broken.