Using the pronoun in this sentence would make it sound unnatural. Besides, this is a question so you would put the pronoun after the verb (*Come scriviamo noi il suo cognome?)

yes. some words ending in 'e' are masculine and some are feminine. In this case it's masculine, so when it's plural it becomes i cognomi

Come si scrive = how does one write (this word, the famous person's name...)

Come scriviamo = how do we write (this word, the famous person's name...)

Come ci scriviamo = how do we write (to one another, for one another...)

Pronouns are pretty complex because they rarely follow one single rule. Sometimes just getting used to the way they work is more effective than memorizing lists of rules.

In this case "si" is not a reflexive pronoun (eg "lui si lava"; "lei si veste") but an indefinite pronoun which means "one", "you" (ex "qui non si parla di politica", one does not talk about politics here).

You would need to rework the sentence. Suo does not carry any info about the gender of the possessor.
Ex. Come scriviamo il cognome di Maria?

I would like to know this too

Well, there are countless situations where the spelling of a word can be difficult, especially with names. English speaking countries have spelling bee contests for that reason :-)

Generally, when using possesive adjectives you need to use the article unless refering to direct family.

Because cognome is a masculine word.
"Suo/sua/suoi/sue" refers to the gender of the things that is possessed, not to the gender of the possessor.

Only context will tell.
The gender of Suo/sua/suoi/sue matches the thing possessed, not the owner

All possessive noun phrases must have the definite article, unless it's a singular close family member.