My question is that Chrome uses standard out stream, to let 3rd party apps to know what happening in the browser?
Another question that Chrome has live log, lately visited websites entered urls etc.
I want to integrate Chrome with a new AIR 2 application, thats why I want to find some solution to connect and interact with the browser.

I doubt it, considering how passive and invasive that is, it seems like it would be considered a security issue. If it did, something the user didn't know was installed on their computer could easily be tracked, right? :P
I'd suggest making an NPAPI extension that can follow the events and output it to your AIR app.