Recently had electrical problems and not sure if affected Remote engine start. New 2006 Rav4.

When I remote engine start, it:
Turns on engine
Doors remain locked
I unlock doors, engine stops.
I have to re-start the car/engine with key.

Is this correct?

Thats the way my daughters Ford vans works. You have to put the key in and turn on so you can put it in gear and turn the steering wheel.

Did you check with your dealership?

Somethings messed up if it shuts off just by unlocking it.