Air Dolomiti taking over? It's a rumor at the moment, not more. When LHI started they first wanted Air Dolomiti to run the business. Some of the pilots even went to do an Airbus typerating. So I can understand that they are eager to get into LHI again. But I personally don't think it will happen soon.
It is not yet really clear to the public why the Air Dolomiti CEO has taken over at LHI, but I guess that LH mainline was not happy with the current managment and wanted someone with thorough knowledge of the Italian market.
Training at LHI is taking place. You will be getting an Airbus rating. What else do you want?