Do you have a static IP ? If not then you will need to setup an account at
http://www.dyndns.com so that your pc will report everytime your ip changes. Even if you do have a static ip I would recommend setting up an account as it will make receiving emails easier using a dns name instead of using an ip. I would recommend removing the router from your setup as it will make things a bit easier and you wont have to play around with forwarding ports.
My setup is like this :
DSL Modem >>>>SME Server >>>>Switch >>>Computers,wireless router
I have had this setup and working for the past 7 years, running on a DSL connection and I dont have any issues with blocked ports. The problem that I do have and that you may have is that because I dont have a static ip many email servers block emails coming me because my ip comes from a known DHCP range of IPs. So keep this in mind if you are not static you may have issues with people receiving what you send.
Removing the verizon router will make things easier then you will need to bridge the modem so that your sme server will get the external ip address making it easier for you to send/receive emails using the domain name you get from dyndns. This way every time your ip changes its instantly reported to dyndns. You can contact verizon and they should be able to walk you thru how to bridge the modem. They may give you some flack and may not want to help you (Everytime I get a new modem I go thru this with my isp and normally on my 2nd or 3rd call I get someone that will help me). You should also be able to find this info online by googling your modem model if the wont help you.
Hope this helps.