Are engines automatically stopped in an emergency landing?
There is no automatic shutdown. However, shutting down the engines is part of the evacuation checklist done by the flight crew.

It's happened before that the controls link was severed and an engine could not be shutdown, such as in Qantas Flight 32.


  Upon landing, the crew were unable to shut down the No. 1 engine, which had to be doused by emergency crews until flameout was achieved.


Normally the captain would then instruct the cabin crew to not use that side of the plane. Same thing happens when there is a fire on one side, in that case the captain would also steer the plane so that the good side is upwind.

Aircraft are tested to ensure all occupants can evacuate with 50% of the emergency exits not working (glass half-full: 50% of the doors working). See: How are evacuation tests made as realistic as possible?

About how long it takes for the engine to stop producing thrust, it's about 5 seconds. If there is no or little thrust, there is no suction to worry about. Also note that figure is for a spool down from full thrust (not from idle thrust, which is to be expected after landing).

(Source: Jet Transport Performance Methods)