That test simply means their port 6346 is open for sending outward messages. However does not mean the port is open for receiving inward messages (that we don't know yet).

Yes, as many details as possible. Otherwise it's difficult to ascertain what the problem might be.

Depends on version of LW they are using. Java 6 is required for LW 5. 64-bit java best in that case.
For other versions of LW 4, it depends on their set up. One version of java sometimes runs better on some people's system than others. Java 1.5_10 I think was the best of the Java 5 versions. Not sure about Java 6 (though have heard of some problems with Java 6, possibly related to particular Java 6 release version).

For connection problems, this is generally the best place to start: If LW Will Not Connect (click on the blue link)

If they upgraded from LW 4 -> 5, then deletion of LW preferences folder might help greatly.