2.77a - Keep foot on the ground with anchor OR do not allow any vertex below certain world Z I am animating a human character. It is bit irritating that I need to go and adjust its feet for every animation I do.

Is there a way, through a constraint, to plant a bone at certain LocRot and not alowed to be moved by the IK? (Actually cannot work well with IK when I think about it).

Or rather, tell the rig that no vertex can cross 0 of Z. That will do it even better.
well, you can place a bone(anchor) in the desired location on the floor, then add a copy location and copy rotation constraints to the feet targetint the bone(anchor). then when the influence of the constraints is 1.0 the foot will be stick to the floor(the anchor bone position/rotation). you will have to keyframe the influence value that can be easy adding a driver for it.