As a father to a one year old, I think there are some key issues you need to consider when travelling with a kid that age. Transport wise having a car will give you some freedom when sight seeing, but for the long hauls, I would recommend the train. I know all kids are different, but based on our experience, trains are by far the best option for longer trips with a kid that has just learned to walk. Our son really doesn't like to sit in a car seat for more than 20 minutes, but you daughter might be different when she reaches that age. However the train just gives you more options to sit and read, walk around, change the kid etc.
The Rhine area between Bingen and Koblenz can actually be seen by train and boat without problem, but it's up to you how you prefer to travel and you could rent a car for a few days if you prefer.
for the area around Aabenraa a car will be a better asset (Aabenraa doesn't even have a train station, you will need to get off at Rødekro and get a bus from there). Haithabu (Hedeby), Ribe, The Wadden Sea and Rømø are all great choices, but if you have any particular interests, let us know. Otherwise I'm sure you friend there can recommend stuff to do and see.
Getting between the two places I would train.
Do note than when travelling on Intercity trains in Denmark, you need to make a 'seat' reservation for your stroller if it can't be folded and brought on as hand luggage. You can only reserve by calling the Danish Railways (www.dsb.dk) on this number +45 70 13 14 15. and a you ned to collect the reservation in one of the machines on the train station. Reservations are not needed on regional trains. I don't know how it works on German trains.
Another thing to consider is that you will be travelling on your kids terms, so you might want to lower your expectations on how much you will be able to see and do. Personally I would put more emphasis on lodging, since I might have to spend more time in our hotel/apartment than I would when travelling without the baby. My suggestion is to find a hotel/apartment in a nice location, preferably with a kitchen and a balcony our a view, so you will actually enjoy being there and don't feel stuck if the little ones decides to take a long nap.