The module 'expo-camera' is an Android project without build variants, and cannot be built.
Please fix the module's configuration in the build.gradle file and sync the project again.
in your build.gradle (and you should), the camera is already included.
And when it comes to
Configuration ‘compile’ is obsolete and has been replaced with ‘implementation’. It will be removed at the end of 2018
it’s just a warning informing us that you’re using Gradle 3 (implementation, api are Gradle 3 interface, compile is Gradle 2). We stick to compile for backwards compatibility.
I don’t know what is the exact issue you’re having with permissions on Android, but you may try following instructions included in my answer here — Can't use permissions API in SDK 29 - #5 by sjchmiela, they may help you.