1. Depends on how well written the virus is. (Most of them are pretty badly written!) However, I doubt the writer cares much about the long term survival of the worm once it's grabbed all your email addresses and then mailed itself to them. 
2. It's not a the intended vector for infecting other machines, but yes, that would work.
3. AV tries to intercept the worm before it gets resident in memory (by checking files when written to the disk, scanning attachments in mail, etc.) but once the worm is running the AV is indeed vulnerable to having it's processes attacked, which is why booting from a clean disk (such as a LiveCD) is such a vital part of malware cleanup.