After a few months of despair in solving the NYGC's networking problem with accessing certain SSL sites, I finally realized that the DSL modem is a router itself!
I had thought it was the problem with the Netgear router and was planning to replace it, with doubts.
What I did, was to configure the Westell 2200 DSL Modem/Router (default ip: 192.168.1.1, username/password: admin/password) to connect to the PPPoE account with our Username/Password from the VerizonDSL ISP instead of doing this from the Netgear Router.
As for Netgear router, I set the "Does your internet connection require a login" option to No (Basic Settings) and manual configure the Internet IP to 192.168.1.47, gateway being the Modem's 192.168.1.1 of course, as well as entering the DNS servers fields manually.
It seems that I have no problem leaving the DHCP servers on both routers enabled, since each station's IP in our network are configured with fixed IP.
Now, the HTTPS sites can be accessed successfully without timeout.
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It appears that after the above configuration, some stations have difficulty accessing some sites such as hotmail and yahoo! Mail.
Therefore, I reconfigured the whole network, so that the Westell 2200 Modem's IP becomes 192.168.2.1 instead of 192.168.1.1, and I set the Netgear router's Internet IP manually to 192.168.2.10 with Gateway: 192.168.2.1 and I entered the DNS Server IPs manually as well on the Netgear router.
So basically I turned the Westell 2200's network of 192.168.2.x to have only 1 client: the Netgear router (192.168.2.10), while from the Netgear router, a separate network of 192.168.1.x is created, just like before all these mayhems began.
Now everything is working, https, yahoo Mail, hotmail...etc.