Lynksys WMP54G network card has problems connecting

The card should support WPA (though probably not WPA2). I have found reception problems can cause more problems with WPA perhaps because logging on takes longer. A simple test for that is to locate the computer near the router (or move the router temporarily) and see if you can log in to the router's user interface.

If so you should try to optimise your reception and avoid interference.

1) raise the router above furniture level
2) Experiment with channels (some will work better or worse depending on your environment)
3) If you can detect strong neighbouring wifi, use a channel 5 stops away from strongest.
4) Relocate cordless phone base or video sender etc.
5) Be prepared to move the computer (or at least turn it so your body is not between the router signal and the wireless adapter's antenna.