Here are the answers to your questions:
1) Would a webpage built with RadControls for ASP.NET AJAX download faster than a webpage built with RadControls for Silverlight? And/or generate fewer transactions? And/or perform faster once everything's in cache?
This depends pretty much of the on the size of the page, the number of the controls. The transactions will be the same, since they are one level under the UI (so it does not matter if we do them from controls, or from the silver light components). Once everything is the cache, the Asp.Net page could still be loaded faster, while the Silverlight allow faster execution (since they are executed locally, without the wait for the server to respond). These are fundamental things though, I suggest you to take a look at the Asp.Net and Silverlight documentation for more information.
2) Do RadControls for Silverlight offer more funtionalities than RadControls for ASP.NET AJAX? And/or do they look prettier or can be made to look prettier?
I cannot tell you for sure which provides more functionality. The toolsets are too big to be compared, and the technology is completely different (I am not sure if anyone ever could compare them). As for the UI, the Silverlight controls are the pearl - one of the key areas Microsoft focused with Silverlight/WPF is the UI and the presentation layer.
3) Do RadControls for ASP.NET AJAX work better for mobile browsing (faster, more compatible, etc.) than RadControls for Silverlight?
Yes, they work faster. Silverlight for mobile is still in development.
4) Is SEO easier to do with RadControls for ASP.NET AJAX than with RadControls for Silverlight?
5) Are RadControls for Silverlight more secure than RadControls for ASP.NET AJAX?
What kind of security are we talking about? Server side, or client side? In any way, Silver light is safer, it runs in sandbox on the client.
Let us know if you have any other questions.