If you are referring to a person, it means "I will defeat you" or "I will win."

It depends how it is put into ACTIONS/TYPE OF CONVERSATION.Usually it would mean being cut off from an activity,work etc...This also means as in i will cut you with something a knife,scissors or so but surely no one will say this unless you plan to harm that person or cut a rope from a person stuck to it or any other thing.Being cut off from a such thing,activity,work,school,game,friendship(left out/not able to be in what was mentioned)

When trees are fully grown you can cut them down. The meaning is quite similar when referring to people. You bring him down from his high position.