He is a serial adulterer, just like Trump. 
Gingrich was having an affair with wife #2 while married to wife #1. Callista is wife #3. They had a multi-year affair while he was still married to wife #2. 
Her qualifications are is that she is Catholic and that Trump needs to throw a sop to Newt for supporting him. Newt imagined himself in a power job in this WH, but he must have stepped on someone's (Bannon? Kushner?) toes, because it didn't come through. Francis is of zero importance to Trump, because the savvier members of his team  realize that the US bishops mostly ignore Francis, and EWTN, also opposed to most of what Francis stands for, is also a cheerleader for Trump.  American voters are far more influenced by the bishops and right-wing Catholics media like EWTN than they are by the pope.