My mother has a 2007 Camry with a 2.4l engine and an automatic transmission. A few days ago we were cleaning out the garage and to get more space we had pushed it out in neutral then started it move it back in, it was a short distance and maybe was on for 15 seconds. it started and sounded normal. The next day she went to go start it and the cranking was super uneven, every 4 cranks there was a fast higher-pitched one. She kept turning it over and after maybe 5 seconds it sputtered like a diesel starting on a cold day just once and went back to cranking. It did that same thing a few times. It started up very rough and had white smoke coming from the exhaust. It drove away still smoking unusually. It has been running and sounding perfectly normal from then on. Is this something to get checked out? This car was bought from a dealer and always gone to the dealer for all services. Never abused and only has 167k.
Yeah, it's wise to check out sudden odd behavior, whether it be by car, animal or human - just to be safe, check oil, coolant, check for codes on the OBDII. Your Camry may have thought it was being disowned.
I don't think that this has anything to do with pushing it out of the garage or the transmission. They are correlated but not causal.
From your description, it's probably a blown (or in process degrading) head gasket. You can test for that yourself. Scotty shows you how:
https://www.youtube.com/watch?v=v0U8Iyf0kk4
The prudent thing would be also to scan with a decent scanner (Blue Driver, MX Plus, Innova) for DTCs (Diagnostic Trouble Codes) and Enhanced Diagnostics and Live Data. That might give you some information.
As for "dealership care," there's no guarantee that they do a decent or better job than a good independent shop OR someone who does DIY work. Dealerships often don't do as good a job. (See the recent posts about oil change overfills, as one of many examples).