Firstly, there is no problem running AdAware with AVG. I assume you mean the free AdAware from Lavasoft?
Now there are 2 AVGs - one is AVG anti-virus which puts an icon in your task bar and is continually running. The other is anti-spyware and used to be called Ewido. It would be helpful to know which AVG you are referrring to.
What is AVG doing with the trojan horses it detects? If it is putting them in the virus vault are you emptying the vault?
I suggest you reboot in safe mode, with system restore turned off, and run all your anti-spyware again - spybot, AdAware and so on.
I also like another free program (I am very belt and braces) called A2. It may be worth downloading first and running in safe mode:
http://www.emsisoft.com/en/software/free/