As I've been fighting a Dell with the same problem for over a week now the Drivers for the WiFi Install but the WiFi Doesn't work at all though the Blue Tooth does and that is part of the same Card. :D
The finial choice that I have come up with is simply that the WiFi Card is now dead and gone to Silicon Heaven or worse still the M'Board has died part way though it passes every test that I can throw at it except to actually work for longer than 10 minutes after the rebuild of the OS or a restart and then the entire system slows to a crawl that would make a 286 running XP look fast. It takes 15 minutes to reboot if it will actually close Windows down and there is no difference between XP and Vista in speed terms of the system. So I'm probably very stale with this type of problem at the moment though I've been asking for a big piece of earthmoving equipment to park on the Dell.
As for connecting the Aerial up this should go to the connection shown in the Manual though in Theory at least it shouldn't make a difference as it is only the Aerial and doesn't have any Voltage entering through these connections. Don't worry about the stain this is probably a product of Manufacturing or maybe even the way that the actual Circuit Board is made when Etching the Tracks on it.
As for telling if the WiFi Card is powered if it is showing in the Device Manager it's getting power though it may not be sufficient to actually run the card so it may be getting the correct Voltage but at a lower Amperage which is the hard thing to check. If you have Access to a Volt Meter or Multi Meter you can check the voltage where the WiFi Card Plugs In but you need to be extremely careful and [b]Do Not[/b] touch the probes together when measuring the Voltage.
Really to go any further you need a CRO to check the output of the WiFi Card and signal traces internally to the computer and as most repair shops don't have the necessary test equipment or even the knowledge on how to use tools like that it's unlikely that you'll have access to this type of test equipment. But don't sell the Humble Multi Meter short they are an extremely useful tool that are a [b]Must Have[/b] bit of Equipment and I've used a Multimeter previously to fault find a Circuit Board that was supposedly working properly according to the CRO Traces at least.
Col