Could men give women trichomoniasis?

Yes. Trichomoniasis is a sexually transmitted disease from a parasite called Trichomonas Vaginalis. Most often women become infected by penis to vaginal contact or vaginal to vaginal contact. It reportedly can infect the mouth & anus but can NOT survive in those areas.
Yes. men may give women tichomoniasis and vice versa. In fact, many trichomonas-infected men may be asymptomatic and unaware. Nonetheless, see Doc timely if in doubt. Best wish ...