Yes, unlike the packages through Disney Travel, DVC does not require a ticket purchase for the Dining Plan. Be sure to inform Member Services you want the dining plan a few days before your arrival at the resort, it will be attached to your room reservation, and you will pay at check-in.
You must tell MS that you want the DDP before you arrive. It cannot be added when you check in. If you have annual passes, you might consider getting the Disney Dining Experience. You get 20% off at most restaurants. We tend to eat too much when we are on the Dining Plan. We feel like we can save money by getting DDE instead. If you enjoy having appetizers, entrees, and desserts at each meal, then the dining plan is the way to go.
Yes, you can add the dining plan and you would pay for it at check-in. You want to call MS though and add it to your reservation so it's noted on the reservation when you check-in.