Sorry, CoreOS is Linux as well, but in my mind it's enough of a hyper-specialised immutable auto-updatable container-specific version of Linux that it warrants a separate category when talking about Docker.
I don't think so. That's what Jeffery Snover is working on in Server 2016 with Windows nano server.
Unless something has changed since the last time I checked, The WindowsServerCore docker image was not generally available yet and requires server2016 (I think it was TP6 the last time I checked)
Docker, to my knowledge, is still exclusively Linux flavors. (Though I'm happy to be corrected if someone knows more than me)
Docker images still aren't generally available, but you can now run Windows Container Images based on the NanoServer docker image (and WindowsServerCore image if you replace nanoserver with windowsservercore in their image URL in the docs below) on Windows 10 (insiders build)[0].