It's possible he doesn't have a router or switch, there are devices such as offered by my ISP called "dual connectors" which simply plug into the modem and allow 2 computers to connect to it.
He probably has a firewall or something.
Of course. If you do not allow the program access to the Internet through your firewall or whatever, the server list won't be able to query the master server, and therefore you have no serverlist.
This isn't a router problem, you do NOT need to initiate direct connection to the server browser, and therefore you do not need to open ports for it. I don't have any of the MTA ports open and the server browser works fine for me.