>> "1) 1st Dec, ta.IsGenuineEx(15, 0, true); executed/activated."
It's hard for you to know that from the return results. Hence, you not giving us the full picture. If your customer activated your app 3 days prior then it doesn't matter that you called TA_IsGenuineEx() on the 1st of Dec (because only 3 days have elapsed, not 15).
>> "1) on 10th Dec, will ta.IsGenuineEx() connects to lime server again or it happens only on 15/16 th Dec ?"
No, it would never contact the servers on the 10th. With either setting.
>> "2) Next connection happens on 30/31th Dec or 10/11th Jan or 15/16th Jan ?"
Again, it depends. When was the last time TurboActivate contacted the servers? This is a different question than when the last time you called the function TA_IsGenuineEx().
It's a simple formula (that I've written 3 times already and is described in detail in the documentation):
Last Time Contacted + X. Where X is the nDaysBetweenChecks variable.