Is it hard on a car to run gasoline through it that has fuel stabilizers mixed in?

 

No, but the purpose of a fuel stabilizer would be to keep the gasoline from going "bad" when not in use for extended periods of time.

If you're planning on parking the car for more than a month, that's when a stabilizer would be best utilized. Any less than that and it wouldn't really be necessary.

 

The car won't be driven but once every two weeks. Instead of putting a full tank in and adding stabilizer would it be better to just keep around 1/4 tank in there?

 

Full tank and no stabilizer. Condensation on a partial tank is the result.

 

Fuel is good for at least 6 months without stabilizer.