Yes.
No
Yes, it is possible.
Yes, especially in Volkswagen cars.
It could possibly
Possibly. Replace the fuel filter to see if that solves your problems.
Smaller tires should not cause the check engine light to come on. Smaller tires will cause the speedometer to be off.
it is probably throwing off a sensor, like the mass airflow or map sensor and causing the check engine light to come on
No, low gas cannot cause a Check Engine light to turn on. There would have to be a problem within the engine or the engine wiring for that light to come on.
Yes it can.
If the fuel pressure is too low that might cause the service engine light to come on and , since there would not be enough fuel pressure, might also cause a sluggish response. That could be caused by a clogged fuel filter also.
Running out of gas my certainly be a cause for the check engine light to come on. It could also be an assortment of other issues.
No, it will not.