Or put it differently: will LimeLM try to connect to the servers if it was activated offline?
If you use IsGenuineEx(), then yes it will. Even with offline activations it will try every N days (the difference is, with offline activations, after N days the user will still be able to use your app if it could never connect to the servers).
No personally identifiable information is ever sent to LimeLM.
Does that make sense?