How to reply to students' emails that show anger about their mark?

When dealing with angry students, your number one priority should be to de-escalate the situation and bring it back to a more factual ground. That means you want to:

• Stay calm. Even if you are getting agitated yourself, you should never let it show through in your answer (easier via email than personally, but the same principle applies independently of the medium). I tend to opt for a more formal tone than what I usually use, mostly because I have found more formal phrasing harder to be misunderstood as dismissive (your milage may vary).
• Provide short, to the point, answers. Answer the student's query or complaint politely, but don't get pre-emptively defensive and don't overly detailedly explain your reasoning. The more you write or say, the more an unhappy student will find to be upset about. There is a time and place to give a student detailed feedback, but when they are currently raging isn't it.
• Stay with the facts. Scrub all subjectivity from your own answer. You want to bring the conversation to a place where you are talking about specific assessment results (exam questions, assignment tasks, etc.) rather than "I felt I did better", and the more you are using subjective statements in your own answer the more the student will feel validated that his own subjective understanding of the situation is as valid as yours.
• Require the student to provide concrete, factual arguments. Once you bring the discussion to a more factual level, you should ask the student why, specifically, they disagree with their grade. Are there specific assessment results they feel were wrongly graded? Are they unhappy with how you aggregated the assessments? Once you are discussing on this level, it becomes easier to both, convince a student that their grade is correct according to the framework, and see cases where your framework is indeed unfair (or can at least be perceived as such).
• Answer, but don't answer too quickly. Especially when receiving complaints via email: avoid the temptation to immediately answer questions by upset students. Give it a day, this will allow both you and the student a chance to calm down. A little delay in your interaction will also motivate the student to think more carefully about what they are actually writing, since a non-question like "How could I get this mark, when my colleague got higher" just means longer delay for them.
• Remember that student's don't have to agree with their grade. At the end of the day, not all students will be happy with their grade, and even after you have explained your reasoning not all students will be convinced. That's ok. You and the student don't have to agree on the final grade. There is a process in place what, if anything, students can do to further escalate a grade dispute (I don't know if that includes sending a mail to the dean at your department, in mine emailing the dean about a grade dispute would mostly result in a confused dean).

Your first step should be to make sure the mark is accurate before you reply.

I once had a student complain about getting an "F". I looked in my gradebook and saw that the student had a "B"! My first thought was that I had written the wrong grade on the report sent to the registrar's office. I went, with the student, to the registrar's. There was a "B" on the report I had sent. The error had been made at the registrar's office. Of course, the student's grade was changed to "B".

While @xLeitix's answer pretty much sums it up, I would like to add that

"A clever person solves a problem. A wise person avoids it."

Students complain about marks if either

a. Your rubrics/grading criteria are unclear or inconsistently applied

b. The students perceive you as pliant.

Of course, there are those students who will just complain for complaining's sake or with the hopes of you just not wanting the hassle of dealing with them. Those are, in my experience, a very small (albeit vocal) minority. Moreover, since their complaints are rarely justified, you can easily handle them.

If you handle (a) well, then students are much less likely to complain about you, and would greatly appreciate the transparency. I make my rubrics clear-cut and public, and ask students to refer to specific rubric items in their regrade requests. I tell them in advance that I do not grade feelings or intentions, just what actually made it to the assessment paper. Make sure that this grading policy is public on week 1; perhaps even get students to acknowledge reading it (say via an online form).

Students won't perceive you as pliant if you fix (a) and stick to it. I actively encourage students who threaten me with going to a higher authority to follow through on their threat (politely), something along the lines of

"You are well within your rights to take this matter up with the dean, and if you feel like you have been mistreated in any way, then by all means do so".

No one has gone on to complain yet. They know they have no standing.

Given that we are teaching in unusual times, you can (and should!) be lenient when the situation warrants it, as long as you are consistent about this as well. If one student complains about an inconsistency or possible interpretation and you find their argument valid, go back and fix it for everyone. If a student asks for an extension for no particular reason, grant it for everyone.