What is the button with a picture of a snowflake on it?

Vehicle information needed

Check wiring under carpet usually under drivers floor. It seems many short circuits are in this area when it controls something in the rear of vehicle. Best to trace wiring from defroster forward.

We have to see a picture of what you are talking about. Is it the stereo tweeter?

A lot of the time on the rear GM door solenoids the mounting rivets break. Have someone operate the door switch while you look at the linkage rod on either rear door(Door panel removed) Might need to just reconnect the linkage......

check all fuses in fuse panel and under the hood

follow this link for a bit of advice , i'd also remove the coil pack for the rear plugs , making sure you mark the HT leads , dont want them mixed up
3 5 V6 G6 pontiac spark plugs locations

sounds like motors shot remove the cowling so you can see more
join a g6 forum find with google

check the connections going in to the window leads and also with a test light ,check if their is power at the terminals that plug into the window leads. does your car still have factory warranty that the dealer can look into this.

check the wires going to the rear window, there may be one bad..also, the ones that physically connect to the glass come off...there is a repair kit available at autozone but I have found that it doesn't always work...my solution to fixing the wires that connect to the glass are some rosin core solder and a 250 watt soldering gun

There are anywhere from 3 to 4 vent, defrost, temperature control solenoids in the suitcase or under dash area. What it sounds like is that one of your solenoids which is electronic and is gear driven has a tooth gear where several teeth are broken and the gear cannot complete the cycle that is selected from the control panel. If it is the vent you are trying to open or close, then it is the vent solenoid that needs to be replaced. Bet to go on line to u-tube, in your brouser select your vehicle year, make, model and include 'changing vent solenoid'. You will get several video giving exact detail on how to check for a bad solenoid and then how to change it.