Who is collecting the Dancing Stars

There are 2 things that I've noticed the system clock being used for: the Crank-a-kai and Yo-kai Cam.

The Crank-a-kai disappears after you use it, but will come back the next day. If you change the system time to a day during which you've already used the Crank-a-kai, it will be unavailable.

I'm sure you've noticed in Yo-kai Cam that there is a daily challenge. For example, "Snap Bug-like Yo-kai!" or "Snap Yo-kai Rank A!" There are 64 of these challenges, each with their own reward, and they rotate on a 64-day cycle. It's possible to complete the challenge, set the system clock 64 days ahead, and complete the same challenge to get double the rewards.

There is a list of the Yo-kai Cam challenges, in order, on the Yo-kai Watch Wiki.

Yes!

You can obtain rewards from the Post Office by taking a certain number of photos. After taking my first photo, I was instructed to visit the Post Office to claim my reward, and got a Bronze Doll. I have recently taken enough photos to hit milestones for 5 and 10 unique Yo-Kai, and each time was instructed to claim a new reward.

There is also a daily challenge. When you go into Yo-Kai Cam you will see a message at the bottom of the screen telling you which type of Yo-Kai you need to photograph to get the daily prize. This prize is also picked up at the Uptown Post Office.

Lining up the circle can be a bit tricky. One thing I've found that works is getting the alignment where I would think it would be successful and moving closer, while keeping the circle's top align with the forehead. Eventually I would see "OK!" and quickly snap the photo before it changed its mind.