Have certified used 2012 Elantra.Why does car SEEM to do a "sidestep" after i hit a small bump on road doing about 55-60 MPH?

that sounds like a loose connection on the loom plug

Hello Bethany,
I know a bit about cars but being English I don't understand about rear rotors. I am guessing it is what we call brake discs or disc brakes.
Unlikely I would have thought the actual rotors or discs would be worn to a thickness of 6mm (which would be bad news) and perhaps you are referring to the thickness of the friction material?

It is only a relatively small percentage of braking effort that is exerted by the rear brakes so the rate of wear is much lower than the front brakes so it isn't unusual to discover new friction material isn't much thicker than about 8mm, though many are thicker and some are as thick as those at the front.

It is a general rule that is adopted by most brake and car manufacturer that replacement of friction material or disc pads is not required until there is only 3mm left and I have seen manufacturer's recommendations saying 2mm.
With 6mm of friction material still remaining which is two or three times the recommended minimum there would be no grounds to refuse certification.

Take your car to the service shop and the guy in the greasy overalls will ask himself if 6mm will last you until your next visit and if he is honest he will replace only after asking you what sort of mileage you cover. If he is normal he will grab the extra profit with both hands...

you have a bearing defective !!
it need to be replaced to avoid damage other parts $$$

maybe a wire jumped out from the starter check an electrician...

Check the fuel pressure

Do you have a tow hitch on your truck? Does it rattle when driving on bumpy roads? I had a truck same year and make when on a certain road the timing would ****** problem traced to knock sensor.

I would check the front struts. They could be worn out (you didn't give mileage), or simply loose. I would recommend you either check them or have them checked as it could be minor or it could cause damage beyond the problem. One way to have them checked is to take it to someone that does front end alignments. No reputable shop would align a car with defective parts as the alignment wouldn't last. Sorry I couldn't help more, but a test drive would be required.

sounds like shocks

Vibration from about 40-60 mph is usually attributed to wheel balancing, but the onset when you hit a bump seems to indicate play in the steering linkages (tie rods etc) or suspension members and stabiliser bar rubbers but you had that all checked out, that brings us back to the wheel balancing.

You need your brakes replaced all around.