Is Cohen who had a vasectomy allowed to Bless the congregation? Also marry? What happens if he does marry?
A man who has undergone a vasectomy is most likely in the category of Petzua Daka (crushed testicles) who is forbidden to marry a regular Jewess per Devarim 23:2 (although this might depend on the specific medical technique used). (See Shulchan Aruch EH 5:8,10.)
The Talmud (Yevamot 76a) discusses whether a Kohein who is a Petzua Dakah can marry a convert. (Note that generally a Petzua Daka can marry a convert, while a Kohein cannot.) It hinges the question on whether בקדושתיה קאי he retains his holy status. The Rambam (Issurei Biah 16:1-2) rules that he may marry a convert because he has lost his holy status, but he still may not marry a Mamzer. The Raavad argues on the Rambam and permits a Petzua Dakah to marry even a Mamzer. The Shulchan Aruch (EH 5:1) rules like the Rambam, while the Rama and Gra rule like the Raavad.
Chelkat Mechokek (EH 5:1) explains that when the Kohein loses his holy status by becoming a Petzua Daka, he loses it entirely and can now visit cemeteries but can no longer bless the people. Keren Orah (Yevamot 76a sv Hen) rejects this thesis and suggests that they only lose their holy status with respect to forbidden relationships. It would seem Bet Shemuel (EH 5:1) agrees, because he maintains that a Kohein Petzua Dakah retains the prohibition to divorcees and Zonot.
I'll note that the fact that he has a physical deformity does not alone prevent him from blessing the people as the Shulchan Aruch (OC 128:30) rules that only external visible deformities are problematic in this regard.
Ariel correctly points out that many vasectomies are reversible. The Shulchan Aruch (EH 5:4) is clear that if a Petzua Daka's problem is reversed, he reverts back to being a Kosher Jew. That said some (see Shu"t Chatam Sofer EH 1:14) argue that were this to be possible in a given man, then it will turn out he has always been a Kosher Jew. Accordingly, if there is a chance of reversal of the vasectomy, the Kohein Petzua Dakah shouldn't marry a convert or Mamzer lest it be reversible, nor should he marry a regular Jewess lest it not be reversible.
This is what the Rabbis Answeredabout Marrying with a Vasectomy The vasectomy procedure is done on the vas deferens (many, if not most times) inside the body cavity (as opposed to the sexual organ extremities). The person is therefore not considered a “Petzuah Dakah” of the Torah, and may marrry or remain with his wife if they had a proper orthodox marriage – (source: “Chazon Ish” E.H. 12:7). If the vasectomy was done on the outer organs, then if reconstruction surgery repairs the vasectomy damage, he may marry or remain with his wife (if he was married). If the reconstruction surgery is not successful, but the majority of the people who have the reconstruction surgery are able to procreate, then even the person who did not have successful reconstruction surgery is still allowed to marry, or remain with his wife (if they had a proper orthodox marriage) – (source: “Igrot Moshe” by R’ M.Feinstein, E.H. IV 30).