I'll answer my own questions and sponfeed my fellow linux users:
1- To point JAVA_HOME to the JRE included with Android Studio first locate the Android Studio installation folder, then find the /jre directory. That directory's full path is what you need to set JAVA_PATH to (thanks to @TentenPonce for his answer).
On linux, you can set JAVA_HOME by adding this line to your .bashrc or .bash_profile files:
export JAVA_HOME=<Your Android Studio path here>/jre
This file (one or the other) is the same as the one you added ANDROID_HOME to if you were following the React Native Getting Started for Linux. Both are hidden by default and can be found in your home directory. After adding the line you need to reload the terminal so that it can pick up the new environment variable. So type:
source $HOME/.bash_profile
or
source $HOME/.bashrc
and now you can run react-native run-android in that same terminal. Another option is to restart the OS. Other terminals might work differently.
NOTE: for the project to actually run, you need to start an Android emulator in advance, or have a real device connected. The easiest way is to open an already existing Android Studio project and launch the emulator from there, then close Android Studio.
2- Since what react-native run-android appears to do is just this:
cd android && ./gradlew installDebug
You can actually open the nested android project with Android Studio and run it manually. JS changes can be reloaded if you enable live reload in the emulator. Type CTRL + M (CMD + M on MacOS) and select the "Enable live reload" option in the menu that appears (Kudos to @BKO for his answer)