I've heard the thing about sheep and syphilis as well, but I haven't looked into it much. I don't think it's something they just have and you'll get if you fuck 'em since I've been with a couple of different sheep on different occasions and I never contracted anything.
I (personally) have never gotten a disease or UTI from sexual contact with an animal, and I have only used condoms with humans.
I have gotten a few UTIs (and chlamydia once) from sex with humans, but luckily nothing permanent/un-curable.
As
@FreakyFred commented on, it's probably a higher risk if you're sharing a non-human partner with other people.