Scott,
Thanks for the hints. I believe that I have found the way to make everything play nicely together. I've simplified it quite a lot to. Basically, the wars (other than the Red5 one) do not need the beanRefContext.xml for anything. All they need is to load the necessary *-web.xml files. In the end, and because the Spring context seems to be shared among wars, one single load of defaultContext.xml, red5-common.xml and red5-core.xml should suffice since they reconfigure the same beans.
The shared object demo is now broken, but I'm convinced that if I put it on it's own war that it will play along with the other wars.
Note: If the first deployment fails, cleanly stop the server and start it again. It seems that not always it clean the cache or deploys on the same order (not sure yet which).