(A) How would I code it?
Very Carefully (and it would be very system dependent in pure c/c++ (unless the latest standard has networking finally. can't recall))
(B) Can I just connect a USB or HDMI and use the appropriate code?
Theoretically yes, but that's far from the easy way of doing it.
(C) Do they need to be executables instead of just running out of Visual Studio?
Those programs running out of visual studio are executables, but if set to debug mode they've got extra libraries linked in for assisting in debugging. I don't know what support visual studio's debugger has for networking, but considering who's behind it I'd say you could probably do this all from within visual studio.
(D) Can this maybe done over Wi-fi?
Yes, Wifi is an option. Also Ethernet. Or really any other standard network configuration.
(E) Or is this something more complicated like over a VPN?
Well that's one way of doing it, but that's going well out of the way to do this.
(F) Other suggestions?
http://beej.us/guide/bgnet/