I don't think that you should be worried if that is what you are asking. In many immigration petitions, a copy of the I-94 is required when the petition is filed..it is just one document that is usually required as it shows when you entered.
LIkely yes...their dog was on their property and your dog entered without permission. If someone has to pay the bill for damage it would be you. It is the same as if there was a small child in the yard and your dog entered the yard and bit him.