Just want to mention, your SPCA is going to push very hard for you to take the cat to a vet to: a) find out the age and GET THE CAT SPAYED/NEUTERED before it starts creating more unwanted kittens; b) get the cat vaccinated, particularly against rabies; c) get it checked out for other health problems like tapeworms and ringworm; d) just in case it's a lost pet, scan it for a microchip.
I know it's a lot of money and commitment, but I'd make sure the kitty gets shots (for its own sake) and spayed (for the community's sake).
You were asking what you could catch from a dirty kitty... I did once get a mild case of ringworm from a stray who moved in just for the coldest part of a winter. And you know about fleas...
Hope it works out for you and the cat who's adopted you!