Well, I´ll try to do my best to answer your questions:
There are actually two Hapag-Lloyd, the tourist and the logistic branch, both belonging to the Preussag Group, a former steel company which decided to enter the touristic market.
The logistic arm comprises the container line, associated businesses to the container shipping line, a forwarding company and a railway waggon provider.
The touristic group Hapag-Lloyd includes the airline Hapag-Lloyd and the cruise line Hapag-Lloyd. Both have been separated from the logistics company when Preussag bought them but as Hapag-lloyd is an established brand in Germany since the late 70ties there wasno reason to change this.
Inside Preussag the large vacation companies around TUI are vertically integrated with Hapag-Lloyd and the other airlines (Corsair, White Eagle Aviation, NEOS, Britannia) which means that you can but full vacation packages. Still you are able to purchase normal tickets, it is called "Einzelplatzverkauf". Most of the seats are sold with vacation packages, only 20% are only-flight tickets.
The HF (base HAJ) fleet comprises Boeing B737-500 and -800s with the -500s being leased out and A310-200s. All destinations are in in or near Europe, over-seas destinations such as Puerto Plata are no longer served as the A310-300s always had some problems getting there, especially during the summer as far as I know. However Preussag will start talks with Airbus and Boeing in fall this year about an 80-frames order to replace most of the current fleet, both narrow-body and wide-body.
Next thing going to happen is Preussag launching its own low-cost airline with Germania being the partner for this. Within 3 years they want to have up to 50 planes in the air.
Hope I´ve answered most of your questions