Well, the bit about being allowed in mosques is a cultural thing more than a strictly religious thing. In the old days (like, the first days of Islam), prayer spaces were frequently visited by all sorts of animals, from cats to birds to dogs to mice to livestock. Some of the Sunni schools, especially Hanbalis, see dogs as impure so they might not like letting them in, but there’s actually a fascinating history behind the relationship between religious practice and animals in Islam