As far as I know, it doesn't remove tracking cookies... they're not viruses, and Avast is an Anti-virus program.
So, the same applies to the phishing and adware question as well. Keyloggers are usually accompanyied by a virus, so they will most likely be caught.
If you want phishing, adware, etc. protection, either download malwarebytes, or get the Avast Internet Suite which has all of the "extras" that you want.