How do start of the season placements work?

There is a soft reset at the end of each season, which loosely speaking averages your mmr with that of a ~S4 mmr, so those ending with a bronze mmr will be tugged upward, and those with a gold+ mmr will be tugged down, and those with a silverish mmr won't change much. Due to that averaging, even super high rank master+ people will find themselves playing at high gold/plat mmr at first and will need to climb back up (which usually doesn't take too long).

Your 10 placement matches use that soft reset mmr as a starting point, and then your mmr goes up/down as a result of each of those 10 matches, and then once all 10 matches are complete you're placed based on what kind of mmr you have at the end. For most people silver+, this will probably be a lower mmr than they ended the season at, but that's normal.

I don't think so.

yes it does. The formula is out there somewhere, but iirc, it's something along the lines of your mmr + 1200 / 2