Should a postdoc apply for a tenured position?

Under most circumstances (not an exceptionally brilliant candidate or a field that is very difficult to recruit in), if a postdoc applies to what is advertised as a tenured associate professor position, then I would interpret the application as asking whether the university might consider hiring at the assistant professor level instead. That's not unreasonable, since job openings do not always get filled the way the department had in mind. The chances are lower if the department is hoping for something else, but it's not impossible.

I'd recommend being explicit about this. If you are actually trying for tenure straight out of a postdoc, you should say so, but it will be a waste of time unless there are unusual circumstances (and it may make you look arrogant or out of touch). If you would like to be considered for a tenure-track position should the department decide to hire at that level instead, then you should make that clear.

No. Unless you have proved the Riemann hypothesis or cured cancer, you won't be hired into a tenured position straight from a postdoc (in the USA). You should apply for tenure-track assistant professor positions.

I actually do know of examples of postdocs being hired with tenure. They were people with very strong records, though certainly not solving the Riemann hypothesis (or even necessarily future Fields Medal) caliber. However, I don't think applying for the tenured in addition to TT job will increase the probability of this happening. This something that a university will do if they really badly want you and are worried you will get a TT position somewhere "better," not because you asked politely.