Can cats get STI's?

Do they get venereal disease?

Yes they can... you name it they can get it, including Feline Herpes, Feline Leukaemia and Feline AIDS (don't worry, they aren't transmissible to humans). Even if they are spayed, they can pick them up from nasty tomcats who don't care if they're spayed or not. They can also get it from fighting with other cats... a good reason to keep your cat indoors!
My cat suffered from Feline leukaemia and was on treatment for years before she got a huge tumour in her throat (secondary cancer) and I had to have her put to sleep.

They can get FIV which is the feline equivilent of HIV/AIDS

Oooh yes...cats can get those wonderful diseases too. They even get feline H.I.V. and Clamydia. It's crazy but true. So..cats can be nasty and skanky just as well as any human prostitute. My cats not allowed out just so there's no "new babies" or any sexual illness transferred because it can be pretty bad when a cat gets something. She'll be able to get spayed soon, so I won't have to worry about babies at least!!

Yes, but the only one I've heard of is FIV---Feline version of HIV, which is fatal.