I'm thinking of redoing it in Adobe AIR
Not having spent much time with AIR, my personal opinion is that it is best for bringing a webapp to the desktop and provide a shell to it or run your existing flash/flex project on the desktop.
Btw, if you don't know ActionScript, I mean its details, quirks, etc, don't forget to factor in the time it will take googling for answers.
Are there any arguments in favor of WinForms as a cross-platform app?
Is there a cross-platform future for Winforms (e.g., Mono, etc.)?
It's always hard to predict what will happen, but there is at least one project (Plastic SCM) I know of which uses Mono Winforms on Win, Mac and Linux, so it is certainly doable. However, they say they built most of their controls from the ground up (and claim they want to release them as open source, but not sure if or when), so you will need to put in some work to make things look "pretty".
I played with Winforms on non-windows platforms and unfortunately, it isn't exactly "mature" (especially on Mac). So what you get out of the box may or may not be sufficient for your needs.
If you decide a desktop app is not the best way to provide a cross-platform solution, you can always take your business logic written in C# and create either a full-blown webapp with ASP.NET or go with Silverlight, so many other options exist with C#.