I've written an amateur, hobby-level engine, described here: https://mellinoe.wordpress.com/2017/01/18/net-core-game-engi.... This was more of a personal exercise for learning and experimenting than a serious attempt at game design, so the engine isn't that advanced.
For something more professional and polished, there is Xenko, which runs on CoreCLR for some of its non-mobile targets. https://xenko.com/
Also, Unity itself doesn't really run on Mono -- it just has a scripting layer that hosts Mono on top of its C++ core. Perhaps not a useful distinction for you, but I thought I'd mention it.
I don't know. From what I understand, this is still just a research project, and not something we have immediate plans to integrate into .NET Core. If we did that, though, I don't see why .NET Native wouldn't be considered as well.
For something more professional and polished, there is Xenko, which runs on CoreCLR for some of its non-mobile targets. https://xenko.com/
Also, Unity itself doesn't really run on Mono -- it just has a scripting layer that hosts Mono on top of its C++ core. Perhaps not a useful distinction for you, but I thought I'd mention it.
(Note: I work on .NET Core at Microsoft)