No. Airpods do not have storage space for media.
When you take any of the AirPods from the ear, it automatically pauses the music.
You can also double-tap on the stem to invoke Siri, and ask it to play/pause the music.
Yes, you can control the volume from the phone using the phone's volume buttons just like we do in case of any other headphones.
The standard packaging includes, AirPods (in the charging case), Quick start guide and a lightning cable.
Look past their offbeat design and you'll find that Apple AirPods' compact size and high convenience factor are a winning combination.