In Skyrim, when I attack with a weapon enchanted to deal fire damage, does that give experience towards my destruction skill?
If so, does every weapon enchantment provide experience towards a school of magic?
No, dealing damage with a weapon that has an enchantment that uses a Destruction effect will not grant you experience towards your Destruction skill. Additionally, perks in Destruction will not apply to damage dealt using an enchanted weapon.
To test this, you can put a weak destruction enchantment on a weapon and hack away at something for awhile. When finished, check your Destruction exp. It will not have moved.
The following bit, taken from this Reddit post, authored by user solzhe, explains how Destruction exp is gained:
XP is gained based on the base magicka cost of the spell and damage done to an enemy only.
Runes gain the player XP based on damage per second, but will only harm friendlies if the player sets them off intentionally (i.e. they won’t detonate by proximity to a friendly, only an enemy).