@Pokemon Trainer Sarah Don't worry about changing names, Teri can just stick to Allie, or come up with something different.
@Gef Sadly the RP boards here aren't he most active ones here on the forum, but I'must glad I finally got some more inspiration.

Where Natural Order is concerned, I don't entirely have details for organization/specific characters worked out yet, though it is unlikely that the general public knows a ton about the higher ups anyway. Most of what the public sees are the members who legitimately seem to think separation of people and Pokémon is best, and who have no idea what the actual plan is. I might be able to provide more info once I have the details worked out entirely, but with these things not really being public knowledge in-RP anyway, it doesn't make a ton of sense to do so.

To answer the second question, I'don't say it is more dependent on background--I noticed a lot of the trainers you battle in Alola don't have full teams, and that might be because the culture of the region seems so laid back, but I am sure some people are more gung-ho about having more Pokémon than others. For instance, most of Teri and Alan's teams are actually just the rescues/rehabs that have the most attachment to them. Teri has to be a bit more careful with his choices since he leaves the house more frequently, but once the RP is underway, it should be pretty easy to tell that neither brother defaults to battling, and most of their Pokémon have reasons for being taken in to begin with.

As for the third, Teri's not going to be picky about the help he gets, but I'd keep in mind what the actual position of a Trial Captain means, so if they are going to work for Team Hourglass, I'd advise them having a certain schedule, or noting that they have a temp replacement doing their actual job or something.