getting rid of your cat wont help anything if your house is already infested with fleas. Frontline or Advantage with kill the fleas that are on your cat but it sounds to me like you might have a flea problem outside your house.
it's a terribly long process to get rid of the fleas so if you can find a place for your cat and dog to stay for awhile it would make things easier.
bomb the house a few times with different kinds of flea bombs that kills eggs! get a pro bug killer to come spray your yard and around your house regularly like once a month until the flea problem weakens.
use bleach on tiles or any hard flooring in your house (make sure the hard floor wont be harmed by the bleach ofc)
after you bomb you should vacuum and empty the vacuum cleaner far from the house.
it's a long process and it sucks but it's doable. best thing would be to keep your cat inside if you can.