Web services (whether used by Flex or not) must be web-accessible through a URL. They can't be accessible to CF only via mappings. If you don't want to expose the entire CFC as web-accessible, try creating a proxy CFC in a web-accessible place that can then call the methods on your non-web-accessible CFC.
Also, can you provide the URL that Flex is trying to use? Can you access that same url from a web browser (including any requisite "?method=" stuff)? If you can't access from a web browser, I don't think you will be able to from Flex. I think you'll need to configure Apache to support whatever URL you want Flex to use, and it sounds to me like Apache aliases might be the way to do it. Can you provide more detail on what you tried with aliases?
Thank you for the reply Carl
Unfortunately it is a development server on an internal / private network, so there is no public access.
Regardless, it appears that the CF10 vs apache mappings were not the problem themselves. We are able to access the cfc directly, and unless a method is specified, it will show the CFC explorer.
The issue that is still plagung us is that the CFC seems to be including parts of the fusebox framework, though there is no application.cfm application.cfc or other automatically included page. We cannot figure out what is causing this behavior. Anyway, I have a seperate thread for that issue here : http://forums.adobe.com/thread/1427279
I apologize for not closing this thread out