Can this be used to pull a 100lb wagon on private property or would that damage the car because the holes are for another purpose?
The towing capacity for your Corolla should be 1500lbs. The car itself should be able to tow the 100lb trailer plus cargo without too much of an issue. The problem lies in where you are wanting to connect the trailer at. Could you connect it there and tow successfully? Sure. Probably 99 out of a 100 times you'd be just fine. The problem is, it's not really a good idea to just find a hole and use it. As Weather Vane stated in comments, you run a really good chance of distorting the metal there because it probably isn't meant to be used as a towing base.
Bottom line here is, get a proper hitch installed. You won't regret it.