I can equip the pickaxe and hit the ore vein to mine it. I would like to be able to equip the woodcutter's axe and swing at a chopping block (or a tree) to harvest firewood. Does such a mod exists? Is it possible to do something similar for firewood?

Sure, just make all the trees and chopping blocks a container that gives up wood when harvested. Probably could clone the existing chopping block settings to apply to these new containers.

If im not mistaken, you can have your follower chop wood for you.. Though he only will do 2, or 3 times equalling 4 or 6 firewood logs...

The problem with having follower do it for you is the animation takes just as long as if I did it. What I'm after is to eliminate the woodcutting animation, just like how I bypass mining animation by hitting the ore directly.

I will look into making the chopping blocks into containers idea.