Is what we do, avoiding touching food and utensils, entering the bathroom, and using the clothes of our relatives who raise a dog, correct or an exaggeration, and is it permissible for us to use their utensils and facilities, and is it permissible for them to use our utensils and facilities when they visit, taking into account the possibility of impurity being transferred from their dog?

The impurity of a dog is a physical impurity (najāsa ʿayniyya). Whoever touches a dog with a wet hand, or if the dog itself is wet, their hand becomes impure. One should not ask a non-believer about the purity of their hands, clothes, or utensils, because the default assumption is purity until impurity is proven. However, it is preferable to avoid and keep oneself free from using their utensils and clothes until they are washed. The utensils and clothes of non-believers are permissible if their state is unknown, because the default assumption is permissibility and purity. There is no objection to using the utensils, towels, and clothes of non-Muslims, even though it is preferable to avoid them, especially if they are known to directly handle impure substances.