The Avatar Logbook can be used to store and re-hire up to 99 Streetpass, Spotpass, or DLC characters. It doesn't store items or between-level experience. It stores stats and equipped skills, but I'm not sure if it stores unequipped skills. Does anyone know?
Yes, it does store unequipped skills.
I finally managed to test this myself by unequipping a skill on a character that I got from streetpass, updating that character in the Avatar Logbook, getting that character killed, and re-recruiting him from the Avatar Logbook. After re-recruiting him, he still had the unequipped skill.
I should also note that other players' avatars obtained from Streetpass will not have any unequipped skills when you first recruit them.
Sadly, no, they do not. When you get a unit via StreetPass, only the skills they have equipped are transferred, and since they transfer the version of their unit that is registered in their Logbook, it most likely means that only equipped skills are registered.
I haven't gotten enough money to test this between save files, I can't speak with 100% certainty, but it doesn't look like they would transfer.