I am not a master tech...but...here is my two cents...
I too have Avast on my HTC EVO. When I installed it I do not remember being ask to Root. And mine runs fine.
I do know that for certain features of Avast to work, I do know that you need to have Root access, such as the Firewall. All of the other features, such as the Virus Scanner, however, should work just fine.
Question...are you sure that you downloaded the official Avast app from the Google Market or Google Play Store ?
If not, you might have a roge version. The reason for my question is that in my experience, official apps from the Google Play Store typically not ask you to change your App Permission Settings to allow unregistered app.
Just my personal advice, I would recommend completely removing the Avast app you currently have installed, unrooting your phone (unless you have it rooted for another reason other that stated in your post), and then go to the official Google Play Store and downloading the Avast app from there.
Also, if you feel your phone might have been compromised, you may need to do a Factory Reset to completely wipe your phone.