Here is the reply I got from the AdRotate developer:
I'm not familiar with ajax search plugins, but generally for an advert to (re)load you need to reload the page also. Such search plugins do not do that.
So rotation of adverts will be hindered a bit. Once loaded though, AdRotate is finished doing things. So your tab opening or random outputs are not likely something AdRotate does. Perhaps the advert you're using. But logically not AdRotate since there is no AdRotate code running or being called at that point.
Hope this helps.