The details vary a lot based on the hosting tier, but yes we do use docker everywhere. You might be interested in these topics which describe some of the techniques we lean on:
I was interested to know how you manage all these servers with docker. For example, when a new customer signs up, how do you spin off a new server etc. You need something to orchestrate all these servers running docker, and manage updates etc.
I have an ansible playbook that handles spinning up the stuff a new site needs (e.g., redis), updates the yml file (like if a plugin or setting is changed in my config file), bootstraps the image, launches the image, does post migrations after launch, and so on.