If Verizon won't allow you to use port 25 for incoming or outgoing traffic I'm afraid it's game over. You will need to use their SMTP server for Internet communication. You can use Mercury (with MercuryC instead of MercuryE for deliveries, and using MercuryD to collect messages) in your local network if you like, though.
For sake of clearness I'll still comment on the various details.
DNS: The error messages from Windows Live Mail indicate that you probably haven't succeeded to create the new A record (mailserver.mydomain.com). There should be just one MX record for the domain, and it should refer to mailserver.mydomain.com (not @).
[General]
myname: mailserver.mydomain.com <-- better with just the domain here in most cases (mydomain.com)
[Protocols] <--- disable modules that you don't actually use (if you use MercuryX be sure to get the patch)
[MercuryS]
Strict_Relay : 0 <--- strict relaying control is recommended if MercuryS is communicating with the Internet
Server_Port : 587 <--- this should be 25 (even if it sadly won't work in your case)
Alt_Server_Port : 25 <---- this can be 587
Interface : mailserver.mydomain.com <--- in several places, will break functionality altogether, write # at the beginning of the line to comment it out
Session_logging : C:\xampp\MercuryMail\ESSIONS\MERCURYS <--- missing S in path, it seems
[Domains] <--- check Mercury help for this
localhost: localhost <--- seriously, is that the name of your server?
localhost: [127.0.0.1] <--- [192.168.1.7] and public IP should be here
mydomain.com: mailserver.mydomain.com <--- unless you use Netware the first part is always whatever you choose to call your server
[MercuryB] <--- unexpected port selections, but I guess it would work
/Rolf