Bypass to get Play Services and ADB/Root working on an official Android Studio image. Found this trick on an obscure SO comment a long time ago and somehow it still works.

AndroidStudioEmuPlaystoreRoot

Easiest way: You should create a new emulator, and before opening it for the first time, follow these easy steps:

    Locate the directory where the emulator has been created. For example, in Windows go to: C:\Users[user].android\avd[your virtual device folder]

    Open file config.ini with a text editor like Notepad.

    Locate PlayStore.enabled key and change its value to true:

    PlayStore.enabled = true

    Locate image.sysdir.1 entry and change: google_apis => google_apis_playstore. Example:

    image.sysdir.1 = system-images\android-30\google_apis\x86\

    to

    image.sysdir.1 = system-images\android-30\google_apis_playstore\x86\