No petrol's a good place to start!
Driving through flood water would cause the ignition to fail and stall the engine.
get an engine diagnostic scan.
What is your definition of "pulling"? What are you trying to describe?
It might be a broken motor mount or transmission mount.
There are pages and pages of things that can cause a check engine light. You need to have it checked with a scantool. Then the code can be diagnosed and repaired.
It can do far worse, such as destroy your engine block. I wouldn't worry about your heater core, I would worry about your Engine.
This is an indication that there is not sufficient oil in the engine ! Driving could cause the engine to cease, in the worst case it would then be scrap.
air leek
It sure would.
The engine running will cause air to come out of a running engine. The Internal combustion engine sucks in air, mixes it with fuel, ignites it, and then blows the resulting gasses out. Also the fan in front of the radiator moves alot of air. So unless you are coasting down a hill with the engine off, it's supposed to be blowing air while you are driving.
What would cause my VW golf to cut out whist driving
Driving like an absolute doughnut would cause that. Or not checking and toping up with oil and coolant.