Chevrolet Repair: 2006 Chevy Cobalt Suspension and Electrical?

Question
Just bought a 2006 Chevy Cobalt Coupe LS at auction for my sons first car. Got a really good deal but beating myself up that I didn't look at it a little deeper. The main two issues are:

Anytime you get up to 55mph, there is a violent shake that continues until you either go above 65 or back down below 55. it feels like the entire front end, not just the steering wheel. The car had General Brand tires on them that were completely chopped up, so I put a new set of Uniroyal's on. It didn't fix the problem. I don't believe it's the alignment, as the car has no pull whatsoever. I took my hands off the wheel for a good 10 seconds and it took me straight as an arrow. Now, the guy that put the tires on did say that the back-passenger wheel looked as though it had been welded. (It has aluminum wheels) He said this could be the cause.

Also, there is something weird going on with the electrical.
1. Instrument Panel / displays are very hard to read. Dimmer switch does not work.
2. Drivers Side Door Switches: Driver auto window switch works 1 out of every 10 tries, passenger auto window switch works fine, lock/unlock does not work at all, and mirror adjustment does not work.
3. Passenger Side Door Switches: Automatic Window Switch will roll up, but not down. Lock/unlock switch will lock doors but not unlock.
4. Trunk pop will not work.
5. Radio display shows a few random letters/numbers but that's about it. You can change the stations and I believe a person could make sound adjustments if they knew the steps without having to read the display.
6. The RPM Instrument will "twitch" after the engine is cut off and the key removed. It will only last about 30 seconds once the key is removed.

We have checked fuses but didn't find any problems. Any chance these are just individual problems that need to be addressed or does this point to a bigger issue? Thanks in advance for any response and sorry for such a long question!

Answer
Wow. Well you have a lot of thing's going haywire. I Think I may be able to help you with a few questions, but not all. Hope this helps. The wobble may be because the shock absorbers are worn out. The cupping the old tires had,indicates that most likely that is what it is. The technician should have told you this.Not that he is wrong .To be more certain of a smooth ride,there are additional things that need to be checked. Be sure that the new tires tread surface maintains a perfect circle as do each of the rims.Put it up on stands, and by hand, spin each wheel, eyeing those two surfaces for any out of round movement. Many times, personally, after buying 4 new tires, I have inherited this problem. I don't know why, but it seems the world's tire makers can only make 3 out of 4 tires without making a marginal one. Check all suspension components for looseness. The usual offenders here are tie rod ends. Replace whats worn out, and align.Check hub and rear wheel bearings for any roughness while up,by turning wheels and listening and feeling for any out of the ordinary resistance,any side to side play. Tires should be balanced already.Check fuses and replace any that are blown and be sure that fuses arent just missing .Install where missing. Your window motors are good,more than likely.Your son will probably replace radio, if he is any thing like most young men and their first ride, so that should not be an issue.I am not positive but, i think most ,if not all of these other "failing " components go through the BCM (body control module). Find out where it is, and see if the pin connectors are clean and the module's case is not sticky with old soda pop or something similar. Check all the switches that dont seem to be working.Most likely most of your electrical issues are related to weak continuity in them from spilled drinks or fatigue. To explain to you how to test each individual one, isn't going to happen here, but if it was me, I would go to parts house armed with list of switches, and buy them and install,after ECM inspection. You may have other issues, like pass.side lock actuator may be broken, and not switch, or supply voltage is absent,or continuity issues outside the switches, but your best bet would be to take a shot on those switches and hope,at least you have an educated guess,that this fixes most if not all the issues, and saves you,big time in the end. Good luck.