As the translation of "how do you write your name", I find it hard to wrap my head around using the third person conjugation. Can I say "come si scrivi il tuo nome" instead? I would appreciate the reasoning behind it, or simply a confirmation: it's just idiomatic?
If I understand correctly, you ask why this phrase uses the third person conjugation instead of the second person conjugation even when you address someone directly. The answer is simple: because the subject in this phrase is not the person, whom you ask, it's the name.
Come si scrive il tuo nome? literally means How is your name written?
(or, totally ungrammatical in English, but slightly closer to the Italian structure and, maybe, easier for you to understand: How your name writes itself?)
The subject of this sentence is "your name" and the verb, accordingly, requires the third person conjugation.
Can you say it using the second person conjugation (that is, Come (tu) scrivi il tuo nome?)? Usually, no. Italian is more impersonal than English (and many other languages), so we don't ask a person "How do you - you in particular - write your name?" We ask how the name is written, by anybody, on any occasion.
In some rare situations, you could hear "you" used as a subject, but it, probably, would be a situation, when a person is asked to spell (out) their name. So, when it's not enough just to know how the name is usually written, you'd like to know it letter-by-letter. But the sentence structure would be different (because it would require either the imperative mood or a modal verb, expressing a polite request) and the verb would be different as well, so you'd use something like "compitare" instead of "scrivere":
Puoi compitare il tuo nome, per favore?, meaning Can you spell (out) your name, please?