Do you know what part I'll need to replace switch?Yes, car shops will replace the part.
Its a very easy job.
As for the shifter- I suspect a dirty switch.
I was told that it is extremely bad to completely strip the transmission fluid and replace with brand new, I believe the story is itll mess up the internal part of the transmission . Oddly enough, I also heard you should mix old with new fluid, so I'm lost.Very easy to replace the purge valve for the "No start after gas" issue - Part number: ACDELCO 12690512
"Transmission Hot - Idle Engine" could be too much transmission fluid added to the vehicle (or too little) . If the whole vehicle isn't overheating - Check the transmission fluid first. If I were in your position , I would drain and replace the fluid anyway since the vehicle is new to you and you don't know what's in there and for how long.
As for not starting when in park .. when you start the car on dash does the car show that it's in park (Box around the 'P') or not .. If no Box around the 'P' look at this video.
You can also try to start the car in Neutral.
If the 'P' has the box then there is no problem with the switch / wiring and it's probably a bad ignition or starter relay see the following video:
Hope these help.