is there a way to animate hair

making hair collide with geometry and using gravity?

Yes just apply a softbody to the particle system (not the emmiter object !), turn on collision for the objects you want to interact with as well as the particle system (also the self collision in the softbody settings), and use fields like wind etc. to get additional effects

Ok, Vertex Pusher, (I have to question this).
I create an outliner window so I can see all the object I can select in my scene.
I select the cube and assign a particle system to it.

There is no way to select only the particle system without selecting the emiiter. you have to select the emitter to turn on softbody.