Update: as Per @RonnieOverby's comments, it looks like developments are afoot to allow Linux containers to run natively on Windows. Looks like it'll be Hyper-V based running in isolation mode but should let you manage the whole lot seamlessly. I'm not entirely sure its Linux running on Windows, seems more like Hyper-V running linux and supplying hand-waving and trickery to make it all seamless. Something to consider as a hosting option though.
You can't run Linux based containers - these rely on the Linux kernel provided by the (Linux) docker host. You would run Windows containers of either the Hyper-V flavour (where the container runs isolated in an 'optimised' VM) or what is effectively docker on Windows. The running container would have to be one built for Windows, there are some on the docker hub - the nano server base image one for instance.
Windows has incorporated its version of the OS Gubbins (namespacing and so on) that makes Docker on Linux work - so you get the similar (the same?) experiance but the underlying implementation is different. The Hyper-V flavour has its own slant to this but is a variation on the same theme.
.NET Core could be put together for either flavour - there is already a linux based docker image for this (in several spins) and if there isn't an official Windows one you'd probably be able to make it or it may emerge at some point.
I don't know about the Win32 stuff - I suppose it would depend on what is available in the base images provided by MS, these appear to be largely 2016 Server-Core / Nano-Server based so if your app can can be made to run on something like that then you are likely in luck. There is a stock .net 4.6 one and a 3.5 on the hub along side a SQL express image and IIS one - so there are native moving parts in there.
The Windows ones won't run on Linux.
I don't know how licensing works for any of it.
All the details are here