Hey Howard, Welcome to the forums.
Question 1, Yes Avast works with I.E. 7 and windows Mail just fine. I'm using Vista Home Premium with Avast and have had no problems.
2. My connection is always on through a cable modem, again no problems
3. Here's what ALWIL has to say about Network Shield, "This module provides protection against known Internet worms/attacks. It analyses all network traffic and scans it for malicious contents."
If your using a firewall (which you should be using at least Windows firewall) You'll rarely if ever hear from Network Shield.