Answer:
He's small but he can climb a tower.
Deep, deep, do they go. Spreading out as they go. Never needing any air. They are sometimes as fine as hair
What comes in many varieties and can't be seen or touched, but it often makes you move?
What has no hands but might knock on your door, and if it does you better open up?
War fought with pieces on a board.
Though not a plant, has leaves. Though not a beast, has spine. Though many wouldn't need this thing, It's more valuable than wine.