F# runs on .Net Core, which in turn runs near perfectly on Linux (incl Alpine). The runtime is about 50MB these days, so coupled with alpine or a minimal ubuntu server, you can squeeze a lot out of .net. F# is slightly slower than C# in some ways but in most cases you won't notice it and it brings a lot more to the table. I honestly wish most C# devs would start using F# for everything by default except the GUI layer, which C# makes more sense. If you are doing purely api/backend stuff, F# can be sharp knife you are looking for on linux.
In theory no, as F# runs perfectly fine on Linux. In practice all real world uses of F# in paying jobs I've seen has been Windows. The big advantage however is that there are probably at least an order of magnitude more F# jobs than Ocaml jobs.
We hire F# dev. You can work on whatever OS suits you need as long as you can maintain it. If most of my colleagues are windows users, more and more are switching to tux. (mandatory, I use arch btw). Thank you dotnet core.