Yes, skriver is present tense, so it must be does he write in English. How many letters did he write? would be Hur många brev skrev han?, past tense.
I don't know about ninja verb though, I'd say English is doing things in a roundabout way adding that "do" all over the place just because they don't like the sound of "How many letters writes he?" :-P
In principle flera is several and många is many. If you translate them that way, you can never go wrong. Sometimes it is possible to use many as well though. For instance, we usually say en bok, flera böcker, and in contexts like that, you might as well translate it as one book, many books.
That would be Hur skriver han många brev?
The word order for questions is
- Question word or a phrase that works the same way (if there isn't a question word/similar, this spot is empty)
Here, you want to know "how many letters" he is writing. So hur många brev works the same way a question word would. Like, you could have said 'What does he write?' Vad skriver han? which would be the same structure but only one word before the verb.
If you put just how first, instead of a whole question phrase, the verb that comes next will be the main verb, so you'll be asking "how does he write".