1. I believe it breaks the instagram privacy policy. 2. Instagram saves all the photos and videos we like. 3. We can screen shot the photos we like. 4. Waste of my dropbox server space
1. I haven't seen any statement about that you can't save photos from some API endpoint, so I don't think it is a violation of privacy policy.
2-3. Saving a screenshot of something you like is an option, but that won't give you the right resolution of the image file. Besides, people are most probably liking photos as they scroll, and many at a time (ex: while waiting in a queue). And then, whenever you have spare time, you can see the original sized photos in your Dropbox, as files. So you can do whatever you want with them. That was the main use case I had in mind at the beginning.
4. Yes they do, but you can delete any file from the folder, since it remembers the files it saved, it won't save the same file again. Even if you don't want to sync anymore, you can unlink your account anytime.