First, you can narrow down your subnetmask for Green network to 255.255.255.0 Unless you plan to have more than 254 devices attached on your network.
The gateway for your RED interface is the internal IP address of your Verizon router. If you have RED interface as DHCP client, this should be autoconfigured by DHCP.
The gateway for your client (win 8.1 laptop) is the green NS interface. In your case the Gateway for your laptop is 192.168.2.111
TIP: configure NethServer as DHCP server on the green interface. Then you don’t have to worry about networksettings on your attached devices. They will be configured through DHCP automagicly. Some devices (like printers) make sense to configure manually.
What you need to check if DNS is configured correctly. If you can access websites through IP address but not through domain name, DNS is your first suspect of error.