Is this possible? Also, are staves affected by your skill in a school of magick? Ex., Would a staff of firebolt do more damage if your destruction skill was at a high level?

No you cant craft, enchant or disenchant staves. Staves are affected by your magic skills. The higher the skill in the given tree the more charges it has. Iirc the perks for destruction magic also affect staves in the same school to increase the magic damage from it.

I think you can craft some staves using something attached to the mage's college. But that's a pretty unique exception. Generally speaking, the answer is no, you can't do these things.

True but that is a "unique" situation, and the player is required to do some pretty hefty leg-work to achieve it.

Yeah, you can make a specific 3 staves, but its not via the default enchanting.