The answer to your first question is "yes" and "no." Some puzzles are dependent upon the completion of others, some are dependent only on factors such as population, research levels, and/or villager skills. The puzzles need not be completed in order.

The only way to get to the "end" of the game is to have all of your villagers die, at which point the game is over whether or not you've completed all of the puzzles. All of LDW's games are designed to be open-ended, so you can play any given game as long as you like! The Island Events will continue to happen, which can change the way things are going for your villagers. Once you get tired of your current game, you can do like most people do and play the game again on different settings and/or try different strategies.