The new ZoneAlarm has a "gaming mode" that you can activate within the systray that basically puts in an "auto-allow/auto-deny" mode (you select which) to silence the popup messages that would otherwise cause the system to hang (such as with a fullscreen game).
ZoneAlarm's other annoyance that becomes especially obvious is if you use a packet capture utility like Wireshark - it will constantly attempt DNS lookups of zonelabs.com. That part's solved by getting that IP address yourself and inserting such an entry into your HOSTS file. Course, if it changes, and you can no longer resolve zonelabs.com you'd have to remove the entry temporarily to get the new IP from your DNS.
I second the suggestion to switch to AntiVir. A couple of my (recently departed) dad's coworkers made the switch from Norton to AntiVir and haven't looked back.
And the NAT masquerading router will only typically block unsolicited external connections, it won't prevent applications from "calling home," whereas ZoneAlarm will provided you don't have gaming mode activated.