The FF and IE problem must be a response header problem.. Maybe you could experiment a bit with the HttpHandler code to set the css response header so FF and IE correctly parse .less extensions as .css .. but I think that should be already working.
As for the best practices, if your app is not really big I'd just suggest using the HttpHandler in production too.
It uses the
ASP.NET cache so the compilation should only take place once, and I don't think for a small to medium app the HttpHandler will add too much of an overhead.
Once you reach the point where you want to statically compile your CSS I'd do that as part of my deploy process. I haven't done anything like that in far too long so I hope someone else on the list can help you with that, but I suspect it will involve calling the console compiler .exe at some point :)
As for the Version Control: Never check in the resulting .css files.. Mostly because any change in the .css is not really meaningful to you as far as it's history is concerned, and you could also run into the problem that different versions of dotless could produce slightly different CSS for the same dotless file, making you log a change that's not really a change to VCS.