You are talking about torque, and yes, 3 axis are more than enough, in fact, every possible rotation in 3d can be described as a rotation around a single vector (axis). This is very helpful, because you can compress a rotation into a vector and an angle. The mathematically construct of this is called quaternion.
No, he is talking about angular velocity, which is not a rotation, but a pseudo-vector, and that pseudo-vector is three-dimensional. (For the more mathematically savvy people in the forum: Rotations form a Lie group and angular velocity is an element of the corresponding Lie algebra.)
I think the answer the OP is looking for is that position, velocity, attitude (that one is a rotation) and angular velocity are enough to describe the state of a solid object in mechanics, as he initially thought. Whatever extra things the Earth does have to do with external forces.