Which version of LiveCycle Reader Extensions are you running. If it is an older release then Flash 10 woudl not have existed when it was released.
I am using ES2 or Reader Extensions 9. I have tried both Flash 9 and 10. From investigation on Web, 10 should be fine with ES2 but I downgraded it to version 9 just in case.
Regardless, it doesn't work with either Flash 9 or Flash 10.
This is only a problem when trying to access the user interface. I can access Reader Extensions AdminUI. I can change settings etc so there is no problem with the install or configuration, I just can't load the user interface to actually reader extend a PDF.
One thing I noticed on my production system using LC Reader Extensions 7, in IE there is the Shockwave Flash Object (version 10) under "Add-ons that have been used by Internet Explorer". I do not see this on the development system with ES2.
If I remember correctly there was a patch put out to support Flash 10. It was not available at the time that reader Extensions was released. I think the fix was in SP1. Make sure you clear your cache in the browser or try the UI from a different machine (that has Flash player 9)
Note that my memory isn't what it used to be.