Short answer: go ahead and install whichever Linux distro you like on Hyper-V and go from there.
Longer answers:
Linux works fine on VMs. There aren’t really any caveats. Hyper-V should be fine. It’s been a while since I used it but I remember thinking it was OK. I preferred it to Virtualbox; I think the Virtualbox drivers made some stuff flaky on my machine, but YMMV. I ended up shelling out for VMWare which I’d used at work. Some distros offer cloud images that are tailored for running as VMs, but unless you’re running a cluster with a lot of VMs I don’t think there’s any advantage, any distro will work. There aren’t any significant differences running Linux on a VM from running it on a physical machine.
As to which OS to use for a host, the commonly understood strengths & weaknesses of each OS apply the same as they do in other domains. Windows has better desktop hardware support, Linux tends to be more power-user friendly, etc. It depends on your priorities which you choose. Maybe the biggest factor is that Windows has Hyper-V, whereas Linux has Xen, KVM, and qemu. Either platform can use Virtualbox or VMWare.
P2V and V2P are definitely things. Searching for them online will return tools that will do this. Linux should be rather straightforward to transfer even without a specialized tool, assuming you aren’t using a distro (or distro variant) that is specially built for VMs. dd should work like a charm. It should be possible to do invert the host and guest.
If that sounds like a whole lot of nothing it’s because that’s kind of the way it is with VMs. They just work.