Let say I want to start a new weapon type at lvl 40, and that weapon type is lvl 10 just from bookshelves. Would that weapon type lvl faster as I kill things and turn in quests if i have 3 of its abilities in my action bar instead of 1?
Also you may already know this but if you just keep a skill that you are finding hard to use or not sure about on your skill bar along with four other essentials it levels even if you don't use it.
I often unlock two skills at a time that may only be useable in certain situations under certain roles and pop them along with 3 good ones on the bar and just quest and level them up.
Some level faster than others but very quickly they are ready to at least morph.
Yes, when i start an alt i get one of each of the class skills with the starter 3 skill points and keep them there and always have one of each tree until that skill line is maxed, even if i barely or dont use it, my alt i have found this most useful on is already lvl 35 and over 42 for each class tree and dual wield(weapon of choice)
if you mouse over the skill line experience bar (at the top of the skill tab), it will tell you what contributes xp to that skill line.
There's an Outfitter bundle addon, that lets you swap skill sets on your bars and gear/weapons on your character, you can use it to swap in skills you don't use, before handing in quests. Doing this will level those skills (or skill lines) quite fast.