Here are some comments on your suggestions:

1. Professional mathematical interest in a question or an answer might well be a good necessary condition for MO but it will not make a good sufficient condition.
If the unprofessional content can be cleaned away easily, the posts can and should be edited.
If it cannot, then deletion is a natural way to go.

 Losing joke questions or similar is not a real loss in my opinion.
If serious mathematical content is removed, there is a reason to object.
Jokes should be collected someplace else (if anywhere), and reputation earned from such posts is perhaps reputation that never reflected one's mathematical contributions.

2. If discussion arises on some post or thread and you think it is salvageable, I recommend going and salvaging it.
Edit it to make it better.

 I have experience in looking at colleagues ending up at MO after a Google search and looking only at the text written there, completely ignoring votes and comments.
MO is supposed to be a repository of useful knowledge, and having bad knowledge (e.g. blatantly false theorems) hinders such endeavors.
Therefore bad content should be removed, not just voted down and commented on.
What counts as bad is matter of taste and of never-ending discussion.
I'd say unprofessional stuff is largely bad stuff (although content can be simultaneously interesting both recreationally and professionally).

3. Notions relating to the social aspects of the field of mathematics are always vague.
That is the nature of non-mathematical things.
If I were to conjure up a full definition of socially acceptable jokes in a math talk, I am sure examples would be found soon indicating misjudgments in both directions.

 MO posts are addressed to people online happening to read MO, not just the OP of each question.
Assuming this reader is a professional mathematician in a related field, the text should be sensible and unoffensive.
These readers will keep on coming in no matter how old the post is.
Therefore it is of importance to curate older content as well.

Having a welcoming and accepting social environment is important for mathematics, and that is something I think MO should help build.
This social environment is largely global due to the global nature of our field, and MO is a global actor.
I strongly think that a lot of mathematical talent is wasted due to potential or existing mathematicians feeling unwelcome.
You are free to disagree.

Frankly, I don't think voting is an entirely appropriate approach to these kinds of things.
If three out of a group of four mathematicians agree to make personal jokes about the fourth one, it is not okay.
Yes, this is a true effect even with jokes that may look like harmless silliness to many.
Three losing the ability to make bad jokes is a far smaller loss than one feeling unwelcome.