The required scopes for a given endpoint are in the ESI spec and can be seen in the Swagger UI. For example, for /characters/{character_id}/skills/ you’d need esi-skills.read_skills.v1.
Regarding the callback url it’s a bit tricker since you’re not a web application. I think the proper way of doing it is using Linking and registering a custom protocol e.g. eveauth-app://callback/. (Note: It must begin with eveauth).
There was discussion about this in the EVE Discord’s #3rd-party-dev-and-esi channel on June 6th. SMT’s author said these custom protocols were a pain and it would be easier to do it via a HTTP server on http://localhost, but I’m unsure you can do it the easy way on React Native. 
Good luck!