Do .net applications run on linux?
Are there any free/paid interop libraries available ?
The Mono project provides a standards compliant implementation of the CLR virtual machine component of .Net. They've also reverse engineered a significant number of the framework portion of .Net. You'll have significant issues trying to develop WinForms apps. Mono provides a list of several graphical toolkits you can use: http://www.mono-project.com/Gui_Toolkits (it looks like they actually support WinForms now; though I'm not sure of the extent of that support).
Note that the Mono port of Silverlight, Moonlight, is officially endorsed by Microsoft. So if you can get away with using that, it might be your best shot for cross-platform compatibility.