Can I get physical custody of my children if their father has committed adultery? How?

Defamation is a 'disfavoured cause of action,' in the US, presumably because people believe in a free country they should be free to say anything degrading about anyone and as a matter of public policy, it's a good idea to make it hard for people to successfully accuse others of defamation. If you sued him for defamation, you would (a) probably lose; (b) go to considerable expense both financially and emotionally; and (c) even if you won, could he pay any judgment large enough to pursue? Adultery is something contemporary family courts do not really want to hear about. The personal morality of a spouse or ex-spouse is generally irrelevant to divorce decisions. Exposing children to adult sex, however, is frowned upon. But surely you'd want to confront him with the facts and demand that he stop, rather than rush into court? Find an experienced matrimonial lawyer and discuss the whole matter with him or her, with all the details.

I don't think adultery is the issue. Exposing your child to his sexual activities is a real issue.

Florida is a No Fault State, so typically, adultery is not going to play any part in the judge's decision on anything. However, if he is posting defamatory pictures of you, then it may be possible that those might have some impact, but not enough for any sort of lawsuit. You need to consult with an attorney.

You should refile because the court summons would have expired by now. Given the facts you stated- you have a good case for sole custody.

Defamation is a disfavored tort given constitutional protections for free speech. You might be able to get him prosecuted for stealing and publicizing private images and should be able to get a restraining/injunctive order addressing that. How did you get those "photos" of him, mistress, etc.? Adultery is irrelevant to custody.