1 Yes, your poor cat will very likely catch them, and you might as well!!
2 Yes, the house is likely to get infested, as skin flakes unaviodably embedded in the carpet and other furnishings, will provide food; fleas don't only eat blood.
3 Get a proprietory flea killer for the house remembering to keep pets, fish, etc out of the rooms being treated, get the cat and dog to the vet as soon as possible, and keep treating the carpets for at least 3 weeks.
I once had a stray cat get into the house, and it was infested with mites, the poor thing. It took several months to get them out of the carpets downstairs.