The arrival of that special someone can be a bit unpredictable, so the best thing to do is just to let your villagers continue to make babies. The last puzzle is a bit of an easter egg, and does not depend upon the other puzzles. You can probably start working on it now. If you need help with any of the puzzles, you can look
here.
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.