No stars are required at all. You can also skip some days if you don't want to do them. Or only to the first stars. However you like. First/silver stars will also increase your score in the leader boards.
I think you need all stars to do the final puzzle.
That might be true, I've only completed it once a few years ago. But if you don't want to the more complicated parts, then you want to skip the last one for sure.
I might be mistaken but I believe the final day has only one puzzle, but the 50th star (a.k.a., the second star of December 25th) is given for having solved everything else. I don't think any puzzles are "locked".
You're right. The second star is a freebie for completing the rest.