OK, the door switch has either failed and isn't working properly or has been pushed out (broken) from it's proper mounting position. Many older dryers will run with the door open, as long as the timer is running, so not to worry about that issue.
As far as the dryer seeming to be very hot, it's normal for a dryer to get hot rather quickly to dry clothes. But, with that being said, check the entire exhaust pipe to make sure you don't have any blockage that might be causing a venting problem - you want the vent pipe clear and clean and it should be checked at least once a year to make sure you aren't getting a builidup of lint, which could lead to problems with the dryer, or worse, a potential fire.
Now, onto the door switch. If you open the dryer door, you should be able to see if the push button switch is present, or has been broken or pushed into the dryer cabinet. If it has, you'll need to replace it with a new one.
The switch you need is Part #3406107 and should be under $20. It is fairly easy to remove and replace.
First, turn off the breaker to the dryer and/or unplug it from the power outlet. Then you can remove the top panel, by using a small, sturdy plastic or metal putty knive topush in the retaining clips that hold the top to the main cabinet.
Get down and look at the edge under the top (about an inch from the top surface) and there should be (2) indents near each side of the top that you'll use the putty knive to push in the retaining clips to release the top. Once both are released, you can tip up the top and get at the door switch.
Once you've done that, you can remove the faulty door switch and unplug the connector and put in the new switch and put everything back together.
Here's a line drawing that shows your dryer and the parts list (with the switch you need) to help you with the process.
This will save you quite a bit of money, versus having a service tech come out to do this rather simple repair.
Hope you find this Very Helpful and best regards!