The birds should definitely not be let out, if the cat is in the room with them. Even a lazy cat is a potentially deadly threat to a finch or any bird. You would want to put the cat out of the room and make sure the door stays closed until the birds are safely back in the cage.
As for giving finches, out-of-cage time, it basically requires the same strategy as letting a parrot out, but with a little more patience and extra care to bird-proof the room and keep track of the littles. You will always need to bird-proof the space to the best of your ability before letting a bird out of its cage and allow yourself plenty of time to get them back inside the first time. Most finches will eventually figure out the routine and go back into the cage on their own, but the first couple of times are going to be a learning experience for both of you.
If you don't think that you will be willing or able to let your birds out of the cage, you should plan on getting the biggest cage possible. It will be their whole world, so it needs to be extra large. Opinions vary greatly on how much space a bird really needs if it is 100% cage-bound.
Personally, I don't think they make big enough cages to give birds enough exercise and freedom to fly. A lot of common cage sizes marketed toward small parrots/finches are not even big enough if you DO let your birds out every day. And for small birds you need to be extra careful about bar spacing - bigger cages will tend to have wider spacing which isn't appropriate for finches. You might look into indoor aviaries, if you have enough room. That can be a great option for finches. They will be safely contained and indoor aviaries can be quite large, providing decent space for flight.