Is it common for students to complain directly to deans, or for deans to review faculty's message to students? I am a tenure track assistant professor in a relatively small university (approx. 6000 students).  Without giving too much identifiable details, I'll just way that a student was unhappy with some aspect of the course (e.g. grading scales) and filed a complaint directly to the department chair and the dean of the college without giving me any opportunity to respond.

I certainly believe I didn't do anything wrong.  At least no other student had a problem with the course.  But that's perhaps not important to my question. (Update: The department chair stated over phone conversations that (s)he does not believe I did anything wrong under the extraordinary situation that we and many other schools are facing. My tone, in email messages, is professional but cold. E.g., using "Mr." rather than first name, using "I can..." rather than "I'll be more than happy to...". The Chair said (s)he would not consider this to be an issue. So while it's less than perfect, the Chair cleared of wrongdoing in private)

Surprisingly, the dean of my college got involved directly.  Sending detailed instruction through the chair telling me what to do.  After several rounds of negotiation, I gave the student exactly what this student demanded.  But still, the dean does not like my tone.  My messages to this student now have to be reviewed to make sure I sounded friendly and helpful.

This whole thing strike as rather strange.  My question is how common are these?  Is it common for students to complain to a dean and get their support?  Is it common for a dean to get directly involved like this? 

A bonus question is, as a tenure track person, what should I do?  (Shut up, do what they say, and update my resume, just in case?)
As a tenure-track person who presumably has the goal of making tenure, you must fall in line with what your dean has prescribed as behavior commensurate with one of their department instructors. If you feel the student has violated some part of the institution's enrollment agreement or your syllabus, these violations should be enumerated for your dean. However, if the dean is talking about tone and messaging, then this is likely an issue of your dean feeling like one of their tenure-track folks are not falling in line with what is expected of a tenured professor at your institution. 

This question--and any answer, frankly--is very specific to your institution and your dean's management style, unfortunately. At the end of the day, if you want to make tenure, you need to consider that your dean will be a powerful voice during tenure review. Even if you don't intend to speak to your students in the way that you feel runs counter to your personal understanding of best practices, I think anyone else would recommend that a tenure-track person ought to focus on behavior and strategies that would more likely result in a positive tenure review.

That means, unfortunately, that you must start sanding down parts of your puzzle pieces so that they're fit with the existing puzzle.