Ditto green granite.
The reason the windows firewall turns off is because Zonealarm turns it off as it starts.
If you really want both running, which is highly undesirable, just go into Zonealarm settings and uncheck the box that says it will turn of windows firewall.
The suggestion that you should only run one is correct, but not windows, Zonealarm is a far more effective product.
If it is the free one you are running then the limited support from them may mean you are better off sticking with Windows however.
WRT your Firefox and Thunderbird behaviour, firewalls restrict ingoing and outgoing connections from all programs.
You may have to go into program control in Zonelarm and make sure all the FF and TBird settings are on "allowed" for local and internet access.