Source's use of IE base code is quite limited... I've no idea how to have the defaullt browser open. I've seen a few posts here asking how to stop it, but don't remember if anyone posted the HTML that was doing it.
As for game browser... why? I suppose you could find some way, using lua perhaps, to have them connect to another game server from a link, but I don't imagine any way to have them open the game browser.