Is it common for students to complain directly to deans, or for deans to review faculty's message to students?

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.

Let me stick to answering your questions directly.

Yes it is common for (a few) students to complain. Usually the complaints will go to department heads but, especially in a small place, might go directly to the dean. This might also be the case if the dean has a fairly public presence, as my own dean did.

No, it is pretty uncommon that a dean would want to monitor an individual faculty member directly, especially at so fine a grain as the communication with a particular student. They have a lot of tasks that would normally seem to be much more important. The fact that the dean is intervening directly here seems to imply that they think the situation is especially (important, disturbing, disruptive, ...). The implication is that it is important for you as well.

This can affect your future both locally and more generally. Good relations with students is an important selling point of many institutions, especially smaller ones. Your relationship with students generally can be an important consideration when it comes time for a tenure decision.

I don't have any knowledge of how you acted or whether it was appropriate or not, but the fact of the dean's intervention suggests that you need to take it as an important message. Some interactions with students are entirely inappropriate, even when the student has erred in some way.

If you have a trusted colleague, preferably tenured, I suggest that you review everything with them and see what advice they have for you.