Once cooked, lobster meat will stay fresh about 3 - 4 days. You can tell a lobster is still alive if it is moving its legs, tail, claws or antenna. If a lobster flips its tail vigorously, it is still strong and has survived shipping very well. When cooked, if the tail curls and the tail meat is firm, the lobster was live before cooking. Do not worry about cooking live lobster.
Lobsters have nothing more than a primitive central nervous system.
There is an easy alternative for cooking live lobster though. Lobsters sleep if placed on their backs. Place lobsters in a pot of cold water and set heat on high. As the water heats up, the lobsters will gradually cook while continuing to sleep. If you're stuffing lobsters, you will need to split the lobsters while they are still alive.