What are the words for the different parts of a ticket?

As to the first, the part the customer retains is called a stub (the returned portion of a ticket), and the other part is sometimes known as the counterfoil (though the term can be applied to other things similar to tickets, such as a money order). The second could be tearing or detaching (there isn't a specific term just for tickets, that I know of), and the third is a perforation.

Edit: since the term counterfoil has been found, I decided to go ahead and put it in my answer to make it complete — but the credit goes to user11761.

Counterfoil is the part of the ticket that is retained by the issuing authority.

Once detached from one another, the detached parts are stubs. Generally, the word is used in the context of the half the customer retains: "When you go to the restroom, remember to bring your ticket stub with you, or you may not be allowed back in." If there's a special word for the half the box office retains, I've never heard it.

Edit, because quotations from the OED make everything better: The Oxford English Dictionary finds ticket stub in use by Ellery Queen in The Roman Hat Mystery (1929): "You'll be looking for ticket-stubs. Anything resembling half a ticket." The earliest definition given for stub, from the year 967, is as a synonym for "stump" (of a tree); many or most of the definitions that have evolved since then retain the sense of a small thing that has been severed from a larger thing—like a ticket stub.

Edit 2: Here's a reference from 1887, in a joke that does not seem to have aged very well since then:

"BATH-HOUSE ROBBER: No use lookin' fer anythin' here, Bill. Ticket stub ter one of Joe Cook's lectures, an' a poker chip. Busted drummer from Boston!"

I guess you had to be there.

The customer's part is usually called a stub. (For example, restaurants near cinemas will sometimes advertise a discount "with a ticket stub".) The act of separating the two parts is sometimes called "tearing"; alternatively, the overall act of a gatekeeper being handed your ticket, tearing it, and giving you your part back is sometimes "taking a ticket", same as if you didn't get a part back.

I don't know of a specific word for the part the venue keeps.