Ball and Cat animation

Really new to Blender person here. I have a small project for my class and two questions connected with it, if anyone could give me tips how can I make my project idea come true I would be so grateful!
Here it comes:
I want to animate a cat playing with ball. The ball slides down off a plane first and when it rolls under cat’s paw I want to make him able to push it. How can I do it? If I try to set a ‘rigid body’ modifier on him then his animation goes completely off.
Another question: when my animated cat moves, at some points the mesh stays like it was sticked to a point in the original position. I thought maybe there are some holes in it or something similar causing this effect, but found nothing so far. Any ideas?
Thank you so much for your time in advance!

Well, I managed to make the cat play work properly, but still confused about the mesh behaviour.

You can try to adjust vertices in vertex groups in edit mode or in weight paint mode. Maybe some vertices are connected with no bones (they don’t belong to any vertex group), so they are not influenced at all.