If you choose to fight a visiting monarch, do they not receive troops the next time they visit

In order to get Miis for use in Puzzle Swap, Find Mii and other games in StreetPass Mii Plaza, you must StreetPass with another 3DS owner, which basically means you have to be in wireless range with their system for long enough for the two to realize each others' presence. Once you do this their Mii will appear in the two games.

For Find Mii specifically, a Mii will be no more useful than a wandering hero, dealing 1 damage per hit and bolting for the hills soon after. You have to tag that 3DS multiple times for them to level up to increase their damage output. There is nothing you can do to stop a hero (wandering or Mii) from abandoning battle once they do something, unless they just sworded something to death or won the battle.

Miis you find in your plaza that aren't from StreetPass are those you've seen in other games, like the Miis you play online in Mario Kart 7. They're dead weight.

Mii accessories are earned by completing levels in Find Mii or by redeeming Plaza Tickets from the other downloadable games in StreetPass Mii Plaza. To get all of them you must play through the entire adventure multiple times.

TL;DR: One's fun with the StreetPass Mii Plaza is directly proportional to how many other 3DS's are in your daily stomping grounds. And this isn't even counting the artificial constraints (e.g. some battles in Find Mii are point-blank impassible without a certain colour of hero).

The first time a panel with pink pieces is unlocked, you will receive a free random piece, which can be a pink piece or a regular piece.

Apart from that, meeting other players with those pieces, or being a special employee at Nintendo, there is no other way to unlock them that comes to mind.