Can I legally post a pic of a trespasser on (Facebook for instance) who came onto my property while I was on vacation without my permission? Do not know this person and they did something on my roof but do not know what. The security camera surveillance could not capture what he did since he went above the sight of camera.
Assuming that you are in the US, you can post a picture of another person on the web. There are issues regarding using a person's picture for business purposes without their permission (for example to sell a brand of toothpick) under the right of publicity, but that doesn't seem applicable here. Just posting a picture is not a problem, but adding a caption "This person tried to break into my house" or something like that could land you in legal trouble, where they sue you for defamation.
It isn't totally obvious what constitutes a defamatory statement: it is, approximately "if it tends so to harm the reputation of another as to lower him in the estimation of the community or to deter third persons from associating with him" (Restatement of Torts, Second). Words are not required, so a humiliating picture could count. An opinion or insult can't be the basis of a defamation suit, since such a statement isn't considered by the law to make a true or false statement. The worst possible accusation would be that the person was committing a crime, because that is a per se defamatory statement and the victim does not have to prove that they suffered any loss; in the case of a per quod defamatory statement.
One way to defend against a defamation suit is to prove that the statement is true. Not that you have a reason to think that it was true, but that it is actually true.