Short answer, yes. Medium answer, images your Dockerfile is based on are still run in a Linux environment, even if it's virtualised differently.
Can I run any linux-based container on Windows? Can I run (are there any?) windows-based containers? If so, does it work the other way around: windows container on linux host? Does it somehow use the recently published Linux Subsystem for Windows, or is it completely different compatibility layer? If it is different, doesn't it seem like a waste of effort?
No, on windows you still have to run a Linux vm which the containers will run inside. Meaning all containers actually run on a Linux host. The new Docker for Windows app only abstract away some stuff so it feels easier working with.
> does it work the other way around
No
I don't think, that's correct. To me that's the whole point of having a native Windows / Mac version of docker. From their feature list:
> Faster and more reliable – native development environment using hypervisors built into each operating system. (No more VirtualBox!)
The quoted part is that instead of VirtualBox one can use Hyper-V. In either case, it's handled by docker-machine which runs a GNU/Linux VM with Docker (host) tools installed, and containers are ran on that VM.
I would be surprised if there aren't plans to support WSL (to run Linux-targetting binaries on Windows "natively", thus have "native" Docker containers) but don't think that's available yet.