Yes.

STIs are carried in bodily fluids and in/on the skin of the genitals.

It depends what operation they have had.

A vasectomy is commonly carried out on healthy men who decide they don't want any (more) children. In this case the link between the vas deferens and the testes is removed, so no sperm get into the ejaculate. There will still be the same amount of ejaculate as before and this can carry any disease which is spread by bodily fluid. (EG hepatitis, HIV.)

If the client has had their prostate removed (i.e. due to prostate cancer) then they will most probably no longer ejaculate. In this case the risk is smaller, but skin-based diseases can still be passed on, for example herpes and HPV (genital warts.)

If you bareback or give OWO, there are risks of catching anything contagious, with sperm or without. Nothing is safe, even if you practice safe sex.  Not sure if OWO is part of safe sex practice, and  I'd doubt that.  If a man says, "I got no sperm, I'm safe and you can do without a condom"etc.. Would you believe that?   

Nope 

Wot VC said - ejaculate (semen) is a mixture of sperm and the secretory fluids needed for its delivery.