For question 1: An AirPort Extreme card can connect to any 802.11g or 802.11b wireless network. Brand is irrelevant, as long as it conforms to industry standards. To connect, simply click on the "Radar"-type button in the menu bar, and select the network you wish to join. Holding down Option while clicking will sort the networks by strength.
In order to connect to the Internet, you need to have either a free public WiFi, or a modem with which to connect.
If you were to get Sympatico Wireless, you would still need a modem (they supply it), as that is what would connect to their network, and convert the signal into regular network (which could then be connected to a wireless router).
Having your network as "invisible" really does nothing other than deter the average neighbor from joining. Anyone who is serious about it can detect an invisible network in a few seconds. If you're looking for security, try WPA.
You can select a network to always connect to. This is in the Network preference pane of System Preferences. If you want to see who has connected to your AirPort, you can view the log, though you might not understand most of it