Also, if anyone can help with suggestions on how we can all get around since we need to all travel together in one big group.
Planning a feasible itinerary - and physically moving a group of 13 from city to city with luggage, purses, toddler bags, etc. - that would be a bold and difficult undertaking with a group this size. So I suggest you move everyone and their stuff only twice - once from your arrival airport to your rental home, and once from your rental home to your departure airport. As Jo suggested - ONE travel base, with one "home" for the duration of your stay, is the best way to do this. When you leave your rental, your bags stay home. If it's a GOOD travel base, you will be able to make day trips without all your stuff to many nice destinations.
You won't be able to do your whole wish list. That said, your wish list seems to mirror Rick Steves' suggestions for seeing Germany, which may or may not be the ideal itinerary for everyone. I'm trying to imagine a group of 13 with oldsters and little kids getting around Berlin on the U-Bahn... So I just want to assure you that in Germany, it is no sin to cover less ground than Rick suggests. If you keep a smaller travel circle you will not only endure less hassle - you will likely see more and experience more as well.
The best overall one-base town I see on your list is NUREMBERG. I would attempt to find adjacent rental apartments (or a large home) there. Nuremberg's advantages:
Direct train access from FRA airport (with bags, avoid a change of trains if possible.)
Close enough for a day trip to Munich - and many other easy day trip destinations are nearby: Würzburg - Bamberg - the Open-Air Museum in Bad Windsheim - Rothenburg - Iphofen From a Nuremberg base you can decide as you go which places you will visit and on which days. The VGN Nuremberg transit authority provides cheap day passes for group train travel to these and other places (use Bayern Ticket day passes for Munich.)
An attractive, mostly walkable old-town center with a large zone for pedestrians only; many museums, WW II history, the Imperial Castle (previously mentioned,) local/regional specialties - in other words, a one-place stop that offers easy access to SOME of things that might otherwise draw you to Berlin, Neuschwanstein (which is NOT a genuine castle) and other distant places.
The Middle Rhine Valley ("Bacharach" on your list) is not very far from FRA airport. It is a wonderful place that everyone seems to like, and I feel really bad about leaving it out of the above-suggested plans. So... if FRA airport will be your departure point, I suggest that you leave 2 nights at the end of your Germany stay for a short stay near the airport. FRANKFURT - or better yet MAINZ - would be logical. MAINZ puts you on the Rhine and closer to the scenic part. Let's say you have 9 nights altogether... On the afternoon of day 8, take the train to Mainz and arrive in the evening. Leave bags in your rooms on Day 9; take the train to Bingen, do a cruise, touring a Rhine castle, and visiting Bacharach and another town or two, then return to Mainz for your final night. On Day 10, catch a direct train from Mainz to FRA airport (25 minutes.)