Now that you can desynthesize any item for the class at any level, I doubt anyone is bothering.

Tinker's Calm and Bacon Broth now increase the amount of skill gained (with HQ versions more effective as usual).

Cheapest is going to depend on your local marketboard (or data center marketboard if you don't mind world hopping).

Items 100 levels or more over your skill seem to award the most skill gain on average so focus on those for fastest. Items 50 levels or more below your skill won't award any.

Amount of skill gained past 300 starts dropping even if you're sticking to item at least 100 levels over your skill. It's pretty slow from 350 on.

The note about getting a bonus chance at rare materials if your skill is greater than item level seems to apply to demimateria.