Guess I'll give this a go...
1) How many rivals?
ONE. Pokemon is game about you, the player, and one of the biggest gripes I have about every title from Gen V onwards is that focus is pulled away from you in favour of these bland NPCs they keep shoving in your face. Keep the cast small so that their limited appearances will have meaning. One rival.
2) Gender & 3) Name
I don't see why we shouldn't be able to choose these two things. If there is only going to be one rival I think it all the more important that we get to, in fact. If we're not allowed to choose gender, I'd prefer to see a female rather than a male, simply because it hasn't been done before outside of the opposing protagonist design becoming an NPC.
4) How are they affiliated with you? (How did you meet/know your rival?)
How about having them as an older sibling? That means you'd get to pick their name and gender without it looking too odd, and whilst its cliche and stereotypical, it's Nintendo, so having the "younger sibling needs to catch up to older sibling and prove they're just as good as them" motivation as any.
5) Personality?
Pokemon characters have personalities? That's a new one. Really I think this should be dependant on age, but going with the elder sibling thing, I'd prefer it if they were a confident, supporting type, rather than a dominating bitch with a superiority complex. It would be nice if they had their own insecurities, too - being the eldest can be a lot of pressure in competitive families or depending on parents (who I suspect would be their usual irresponsible selves) and hints of that from time to time would be nice. Make them the kind of rival you would aspire to be like, rather than one you want to beat the shit out of just because.
6) Design?
Something suitably reserved and mature. Let's not have someone with crazy hair or a ridiculous costume, for a change. I've always preferred understated designs, so anything in that sort of vein would be fine. My personal taste in character aesthetics would demand white hair, though, and an androgynous appearance no matter what their gender.
7) What would their team be like?
Bloody good. See below.
8) How are they involved with the plot?
Make them the region's Champion from the get-go. Now you have more than sufficient reason and motive for your own journey - they're your sibling, they're better than you by leaps and bounds from the start, and you just know it's going to be very satisfying to beat them at the end of the game. Make them the end goal.
As for how they'd be involved with the minimalistic and disappointing Legendary-related plot all Pokemon games have...have them be the leader of the evil/misguided organisation as well. Making them the boss as well would be pretty cool; using their influence as Champion, and through their own motivations, they could be central to the game's story. Given that they'd also be your rival and sibling, and would appear in those roles as well in game, they could become quite a complex character. The problem with Pokemon NPCs is that there are too many of them and they don't appear enough. Rival/Champion/Team Boss is three roles, so put that in one character, that's a lot of appearances. That's a lot of time to turn them into something special.
9) Character Development?
The rival should be a fully developed character from the game's beginning; what is important is how this is shown. You can't develop a character when they're absent for most of the game. NPCs don't get character development; they don't show up often enough, and it's an incompatible idea with Pokemon titles, where you travel by yourself. Focus should ALWAYS be on you, the player character. Draw focus away from that, and it ceases to be your journey.
But if they were the game's boss, changing their outlook and motivations for being that boss could be a part of that. It doesn't HAVE to have that stereotypical "you're wrong in doing this and you need to stop" vibe or conclusion to it, either. They could come to that realisation on their own and have control wrested from them by their trusted lieutenant, or they could even ACCOMPLISH their goals, depending on what the mascot Legendary was and what they hoped to achieve by harnessing its power. I mean, as a terrible example, if it was a Time Travel Pokemon like Dialga or Celebi, and they wanted to stop a past event from happening...going back to the past and achieving that, or realising it couldn't be changed, wouldn't be such a bad thing, would it? It doesn't have to be stereotypical good and evil.
10) Their wish/goals of being a Trainer? Do they want to end up as a Trainer?
This would tie into their backstory.
11) How/what do they end up like?
The important thing is not how they end up, but how you end up. Ideally, things would be presented in a manner where you come into your own by beating them. It's a culmination of YOUR journey; the rival has already had theirs and achieved everything they wanted to, and now they've helped you achieve your goals, too.
They don't suffer at your expense by doing this, and there is no need for them to change dramatically as a character. As they don't travel with you, and won't appear too often, they're always going to be nothing more than a supporting character, and that needs to be taken into account. However they end up, I think that needs to be presented in how they end up in relation to you. They're still just an NPC. In my example, having a stronger familial bond and understanding of one another; letting them rely on you a little too (perhaps shown as a result of the plot I was going on about earlier) would be enough.
Maybe have them as a very strong and independent type, who realises that they don't have to do everything by themselves and that they can rely on you for support too. It would depend on their personality (haha) and how that was presented. Whatever flaws they have, let them acknowledge and take steps to correcting them.
