For your first point.
I like your idea about the guild icon. However, wouldn't it be a bit weird to show it above your head? Considering there is no other hud overlays icons on screen?
Tho, it would be nice if it could be a toggleable option for users, in which case they could toggle it on or off from settings? Both Publicly seen and visually seen?
For your second point.
I have seen a lot of suggestions regarding this.
I believe strongly a upgrade in which case is stated here, would make the use of the pokemon storage unit a bit more easily useable.
Especially with a search functionality and a filter, in which case could include index nr, alphabetically or catch date, high-low level etc.
It would be possible, if they were to rework the current functionality of the storage unit itself.
For your third point.
It would be nice if nicknames were only on our visual end. However, I don't know a whole lot about these neety tec wizardly coding stuff thngs.
But, I do not believe they would create a own application just for that, considering you always have to download a newer client from the website.
If they first focused on making their gameclient a standalone client, which you can push updates through, it would be a bit different outcome, I believe.
TL;DR
I think all 3 points in this suggestion could be easily done and could potentially have a better outcome for the overall community.