Started making a humming or vibrating sound, and the color and resolution went bad at the same time. Now when I attempt to turn it on, I get the same vibration sound and nothing else.

Ignitionswitch on steerin colum or the keylock. Nevr go to a dealr..bad techs n doublprice

Probably a wheel bearing

A transmission problem would not make the steering wheel vibrate. Have you ever had the Transfer Case fluid changed? This is required every 50K miles. A qualified transmission technician should evaluate your problem.

may need an e.t.m cleanout and e.t.m breather cleanout most good mechanics can do this. you dont need to replace anything.

If it loud humming noise when you turn it on - touch the side of the TV. If it's vibrating enough for you to feel it - you have problem with color wheel. It could be separate replaceable part or it could be integrated part of optical engine.
If you hear buzzing sound that goes off and then back on couple of times - could be bad lamp (even new one), bad connection between lamp and ballast (lamp power supply), or bad ballast.
If you have to replace optical engine and you out of the warranty - don't bother with repair. It would be not less than $600-$800. Invest in new TV.

Sounds likea bad color wheel to me, or possibly a lamp has blown. If the noise is a humming sound--it is the CW--if you hear a "sound like a sizzle, for a few seconds and then normal, and this repeats a few times--the lamp has blown.

That sounds most like the color wheel,
to change that you need to have some skill in electronics, I would recommend to take it to a shop

Your color wheel needs to be changed. Mine just did the same thing. I changed mine, and the buzzing sound went away, however now my picture washes out with light or yellow colors. The darks look green and purple also.

check the capasitor near the plyback value 250 volts and also the B+ capasitor

If the sound, sounds more like a Hum with some vibration chances are the Cooling fan located behind the lamp is going bad, about $20 to repair, if its a high pitch hum most likely the Color Wheel and if thats the case about $30.
Good luck, my fan just went bad now seeing green dots everywhere, so debating on just junking it out or fixing it, with winter coming on might just breakdown and buy LCD and say the heck with DLP from now on.
Good luck to you..