I don't know what you're trying to do with this story, unless you're asking the questions for the readers to begin with, in which case I'm iffy on whether or not this is plausible, because the questions you pose clearly mean you know that there are a few problems with the story, but you have intentionally left them there. How peculiar.
In any case, I also find it extremely odd that a lot of people would show up for a single Pokemon, because healing Pokemon must be an every-day sort of thing and pretty damn common. So remind me why they all want one little Pokemon? It's just a single Pokemon in a world where there are Pokemon everywhere. Besides, you can get starter Pokemon in a lab or grab a PokeBall and walk in the woods or something for a bit and see a Pidgey or a Rattata or whatever you might find. The point being is that it's weird for that many people to try to get a single Pokemon the Pokemon Center healed. And it's not really the Pokemon Center's right to give away Pokemon so easily, because if the Pokemon was endangered or something, then it would make more sense to release it back into the wild. It got injured, so they healed it. It's as simple as that. If the Pokemon didn't want to be with a trainer, well, I'm pretty sure they would respect its right, because Nurse Joys are sympathetic. Generally speaking, anyway.
The story itself is a bit odd, considering very little was dedicated to thoughts of any sort while the descriptions were also lacking. The very few thoughts imported were disjointed and didn't make too much sense with the context. Actually, most of the story was disjointed. A lot of things just sort of happened, while I mentioned the Pokemon up for adoption thing.
Not sure where you're going with this either. I don't get why you're posing questions, which should be answered rather than you asking readers.