Is saying "drag all politicians out and draw and quarter them" a violation of UCMJ?

A lot of things are being said but not done. Yes, there are provisions imposing restrictions on freedom of speech while subject to the UCMJ. Article 88 of the UCMJ, 10 U.S.C. 888, makes it a crime for a commissioned military officer to use contemptuous words against the President and Congress, among others. The Department of Defense has also expanded this rule to include all military enlisted personnel (DOD Directive 1344.10).

I believe so. Then again, it sounds like a pretty good idea

No. It is free speech and not actionable. One could say "All politicians" or "Republicans/Democrats" should be drawn and quartered, and be fine. To be chargeable it has be shown as contempt towards a specific individual. IE Nancy Pelosi, Mitch McConnel etc.

This could certainly be counseled with an advisory that it COULD lead to UCMJ actions if it continues. Could or would is really objective and up to the charging authority and how they see it and their ability to press and prosecute the case.

Yes, UCMJ Article 88 - Contempt Toward Officials, if you are an commissioned officer. Article 134, General Article for Enlisted.

Yes.

Probably someone from JAG should weigh in on this. But I think no it is not. The three specific elements (with exception) that define a crime which the prosecution must prove beyond a reasonable doubt in order to obtain a conviction: (1) that a crime has actually occurred (actus reus), (2) that the accused intended the crime to happen (mens rea) and (3) and concurrence of the two meaning there is a timely relationship between the first two factors.