Thanks for the suggestions Bjorn. I did update my crossdomain.xml file on both my production machine and my beta machine. I'm not sure if that did anything, but I was able to start test some other situations, including testing with the API beta.
I downloaded and installed the flash player and installed Flash Build 2.6. Unfortunately, due to the extreme cold weather here (or so that's what they are blaming it on), the power outlets in my office seemed to have fried, so things slowed down. (They should be in to fix in the morning).
But with some preliminary tests I did, it appears that if I locate the web folders outside of the inetpub/wwwroot path (e.g. in my profile) and opening the .html, I'm able to access my production services using the API without a security error. But if I copy it to the inetpub, it fails. So I think I have issues with inetpub. I'll look into this once I have my other machine back up.
However, the Flex View App Builder still will not browse to the production machine, even after waiting overnight (so not a timeout issue). Could the 2 issues be connected? Is it better for me to put in a beta tech support request for this, rather than the forum? I did install the Web Adapter, but I may have set it up wrong for our multiple IPs. That's the only guess I have.