My 2010 chevy hhr is stuck in park wont go into gear transmission auto I start the car push down on the brake and cant move the shifter at all cant put it in drive or any gear

there are a number of ideas in this link. Because moving the steering column fixes the issue, I think you have a bad connection under the dash.

https://www.google.com/search?q=2010+Chevy+Silverado+shifter+stuck+in+park

..

Lots of modern vehicles fitted with auto transmission have a safety lock preventing the gear lever being moved unless the ignition is on and the brake pedal applied.

This probably applies to your car and if the system is faulty it will leave the gear lever stuck. Usually it is possible to hear the lock release.

If the transmission end selector lever is accessible, disconnecting the cable or rod from the lever and moving it by hand will provide a clue whether the problem is inside the box or the cable or gear lever.

I have seen in the past a failure of the brake light switch...try unplugging it and by passing it

Brake switch or the release switch for the shifter. Could be wiring, doubtful though.

you have to know if the problem is inside transmission or outside
first under truck unplug shifter cable then try to move the manual lever of transmission near the pan but have hand brake on so the truck won't run over you.
if the problem is inside the transmission could be the parking pawl nut to manual shaft too much end play
if it outside transmission could be trans, cable or linkage
hope it helps

That is a safety item. look in user manual and you will see procedure.

does the clutch push normally or is the linkage broken not disengaging clutch, and then does shifter move around or locked and wont move, loose linkage in shifter or gears are jammed inside transmission and something went, if shifter wont move at all then trans is jammed inside and needs repair.

This time only, your dealer is right. The sound is the Brake to Shift Interlock (BTSI) solenoid releasing the shift lever when you apply the brake. The sound is normal and won't hurt anything but I understand your concern.

have you checked the brake lights for operation ?
if they don't work than it is your brake light switch located on the pedal support bracket.

I had the same problem and once I wiggled the shifter enough to get it out of park then the key stuck in the ignition started.