After talking about it in comments and searching around on the web I think your best bet is to enable Self-Registration in your community:
Once that is enabled your app should contain a link directing new users to this page to register, Luckily, most mobile devices have an easy way to route users to a web page for registration.
Once a user self-registers and gains their password, you'd then want your app to use OAuth to handle authenticating the new user:
The user experience would be to click a login button, this would direct to the community log in page where they can enter their credentials, then the server would respond with the authentication token which you would use in all subsequent rest api calls.
If you want to minimize login requests you can configure how long authentication tokens last for before they need to be refreshed:
There are a couple authentication methods you could try other than username/password:
- User Agent Flow
- Web Server Flow
Once you've done that you should be able to do what ever you want with REST by providing the token. I'm not sure what limits are placed on community users and the rest API. I know your link shows that there are 0 API calls per day but I wasn't able to find a definition for what API calls impact that limit.