mspelled

The server will never realise that you are two different computers connecting. This is because your internal network is private and unable to be seen by the DayZ server. There is no issue with your router - you are able to get to the server correct? Regardless of if one of you is not able to play or not. That means that the server on the other end is rejecting the connection.

A proxy sever will only reroute traffic on port 80 (HTTP) by default (ie a web proxy) - there are specific proxies for different ports by 99% are for HTTP traffic. A VPN will cover off point A & C. Instead of rerouting just web traffic (port 80), the VPN will reroute ALL traffic from your computer to its interface. Have a look at the following article from Microsoft for connecting to a VPN (This guide is for Windows XP however connecting to a VPN from Windows 7 nearly identical): http://support.microsoft.com/kb/314076

Correct. Both of your computers will have a seperate private IP address. This private address identifies you on the network and usually starts out as 192.168.0.XX. Your IP address on the internet is what is know as a public IP address. Your public IP address is assigned by your ISP to your router, which is what the rest of the internet sees. To test simply google "whats my ip address". :) A solution would be to connect to a VPN from one computer (which will route all your internet traffic and give you a new public IP address) and run the others internet traffic through the router as per normal.