Nigel,
There unfortunately doesn't appear to be that much info out there for making responsive map apps with the ArcGIS API. I've modified our Intranet site to be responsive, and it works very nicely. It wasn't terribly difficult to accomplish either.
I haven't messed much with button sizes yet, but this is definitely on my list of things to do in the future.
Unfortunately, I haven't made a mobile-specific app, but as I've said, I'm quite content with the responsive app we now have here. Let me know if you have any questions about how ours works!
--Alan