If I unlock a weapon via Gear Up, do I carry it's level (proficiency, attachments etc) through further prestiges?
A short and simple answer: NO
Every time you prestige you have to level up every weapon again. So if you had level 31 FN P90, after prestige it will be level 1.
You can buy unlocks for weapons when you prestige so that you can use some particular weapon or grenade at the very beginning(for example create a class with an MSR at level 4).