Easiest questions first:
No, you don't need anything other than the .nds-file in order to play.
You do not need to start server.exe in order to play over wifi, just connecting to ted.getmyip.com from the game will be enough.
The problem with not being able to start the server has something to do with some missing C++ runtime libraries or so i'd assume. (or you could say it's my fault for not including the needed ones i suppose)
If you do want to run a server of your own i'd assume installing the .NET Framework (
here) would help. (the current server is not a .NET application, but i think you get the needed files with the framework, and you will need the framework to use the new server anyway, when it's ready)
Progress update:I'm currently rewriting the network code from scratch, probably moving over to UDP instead of TCP, or both, i'll see how things work out.
The server will be rewritten using C#, giving it a gui and such.
Nothing major will be added to the game itself until i get the netcode working properly, but small fixes will be included, so if there's something bugging you, do post a note here so i know about it.
And thanks for all the nice comments. =)