You have to enter your known traveler ID number when you make an airline reservation (and you need to understand that not all airlines participate in the program). If you enter your number when you reserve, and if the airline is a member, then your boarding pass will have an indication that you are pre-check approved on it. You should not have to show anyone your card, and to be honest, I'm not sure if it can get you into the Pre-check line should your airline not participate in the pre-check program. I'm pretty sure it does not.

Global Entry operates a bit differently. Every member gets it, so you just walk to the kiosk and scan your passport and fingerprints. Then answer the questions.

You might need your card if you are checking into a pre-clearance flight in a foreign country

Your status is linked with your passport. If you get a new passport later, don't forget to update your account online with your new passport information. The card which comes by mail is used for land and sea entry. Your Known Traveler Number (for Precheck) can be stored in your frequent flyer profile.

You can now do your Global Entry interview on arrival from any international trip at an airport that offers the service, so I'd definitely check to see if that's a possibility. For the moment, no appointment is required, and the wait time is short (one of our editors did it on Sunday in 5 minutes). This obviously won't help you if you want to use pre-check on your outgoing flight, but it's better than waiting months for an appointment if you live far from a major international airport.

You should check back in your online account, to make sure you actually submitted it and that it shows it is being processed. A friend just told me she waited forever, then found out she hadn't completed the process (eg. Paid her $100).