a) What would be the most common mistakes people make when debugging Ajax?
I'm not quite sure how to answer that one. If you are using a debugger, rather than alert(), then you're heading in the right direction! It's worth spending time with Firebug, and/or the IE Dev toolbar and MS Script Debugger, and Fiddler to figure out what they can do - with the DOM inspectors, debuggers and HTTP listeners, you can get a good handle on what your app is doing, and even modify it on the fly if you need to experiment a bit.
b) Do Ajax technologies expanding outside of Web app development in the future? If so, how?
Several people seem to think so! Adobe's Apollo is heading that way, and various flavours of HTML-based widgets are already here (Yahoo! Widgets, mac Dashboard, Vista gadgets). Firefox is a nice development platform too, I've been told, and version 3 is providing offline storage, apparently. So there's definitely a blurring of the boundaries between web apps and desktop apps.