First, treat the cat with Advantage Flea Treatment to kill the fleas on it. Advantage is sold for cats or dogs based on the animal's weight. You can find it at pet stores, the veterinarian's, and online. This way, the cat won't continue to bring fleas into the house.
Second, examine your hamster for fleas. Check his belly, armpits, neck and tail area, especially. A flea comb works well to check the overall body. Any fleas you find can be drowned in soapy water. (They're not easily squished.)
Clean the hamster cage thoroughly including washing it out with soapy water, rinsing well and drying it off. To get rid of any flea eggs or larvae in your house, vacuum or mop the floors (especially in cracks), vacuum upholstered furniture including under cushions as well as your bed, wash all bedding in hot water. Vacuum the floors daily for several days to catch any newly hatched fleas.
Call a veterinarian that specializes in small animals (rodents). Ask for advice on any flea treatments for hamsters. I doubt that Advantage can be used on small animals like hamsters. Good luck!
You can buy over-the-counter hydrocortisone cream to put on the bites on your arm. Topical Benadryl lotion also helps.