Is a mana dorks tap ability something I can respond to and prevent? For instance, I attack with {Thragtusk}. My opponent declares {Avacyn's Pilgrim} as the blocker. Now, before damage, he taps the pilgrim for mana in the hopes of casting X spell. Can I prevent that at all?
Tapping mana dork for mana is treated as a mana source, just like a land. I don't know of anything at all that can stop a mana source being activated.
Yea unless you remove the ability entirely ({linvala, keeper of silence}) you can do anything about that, you can respond to what the mana is used for though (unless it's a split second spell like {sudden death}).
Like the above posters say, it's treated as a mana source and can't be responded to.
You can, of course, respond to whatever he uses that mana to do. There are a number of ways to make that mana produced go to waste outside of counterspells, such as by ending the turn and exiling the stack with {time stop} or {sundial of the infinite}, or {mana short}, or other elaborate scenarios.