Sometimes yes, sometimes no. They're not the same, but interchangeable in a handful of situation. At its core, flera means "more than one". Depending on the situation, this can mean a few, some or even many. Många on the other hand always means many.
To be on the safe side I've been using "several" for "flera" and "many" for "många".
Not really. Maybe colloquially if your cousins are young you can call them tjejkusiner ('girl cousins) and killkusiner ('boy cousins'), but usually we just use the same word.
This is different from Spanish, which uses "primo" and "prima" for male and female cousins respectively.
same with french and german, and the collective group is masculine too, which I always found a little annoying as I have more girl cousins than boys XD
and what would be the difference between mycker and många? mycket bocker / många kusiner both seem to express countable quantities
många is for countable nouns and mycket for mass nouns. With böcker, you can use either depending on whether you think of them as many individual books or just 'lots of books'.
It's interesting, that there's a distinction between morfar and farfar, but not for the cousin's gender, which I think is more common in languages than the exact relation to one's grandparent.