JSON.NET is the standard library for handling JSON on the .NET platform, and to my knowledge it's available for both .NET Core and for use with F# collections.
I've never used F# on .NET Core, but I'd be surprised if it didn't work.
Yep F#, the language, works all right in .NET Core 2.0. Some libraries, particularly, type providers, not yet, but they are working, on it. And JSON.Net also works fine on .NET Core.
Chiron is what I've been using. I like it because it's simple which means I can understand it. Currently using it with fable elmish architecture, again because it's simple and i understand it.