In what ways can force be applied to a black hole
Aside from gravity, electromagnetic forces can change the momentum of a charged black hole. 

When something falls into a BH momentum is conserved according to every observer, although they may have very different ideas of how that happens. A distant observer will see the new black hole as having some momentum and so moving in some direction relative to the trajectory of the old black hole. It's not really appropriate to describe it as pushing, though. From the object's perspective, nothing special happens as it passes the event horizon. From a distant observers perspective, the gravitational field of the moving object merges with that of the black hole, and the result is the field of a slightly bigger and slightly moving black hole.