Q&A

Idle speed is typically controlled by the computer, idle air control control (IAC) valve and throttle body. A common cause for low idle (and stalling on braking) is a dirty throttle body. Carbon gets coked onto the throttle bore and valve (butterfly) and starts to close off the opening at idle. This typically will not set any warning lights. This will look like a thick, black ,oily goo when you remove the intake air tube. Spraying throttle body cleaner on a clean shop towel and cleaning off the bore and butterfly valve would be the first thing if the throttle body looks dirty. Also make sure your air filter isn't restricted (has it been on the car for 20-30k miles?)

From my understanding, the IAC valve would cause the check engine light to illuminate if something were malfunctioning.

You may also have a bad motor mount if the engine is moving excessively. It's difficult to say without looking at it though. Sort out the low idle first, and then see if you're still getting engine movement

Inspecting the throttle body and air filter would be the first thing I check. hope this helps!