Sounds like a good trip! You have about 5 1/2 travel days within your 23 with all of your various destination. Also factor in your first day which will be overshadowed by jet lag! Make sure to consider both when deciding how many days you'll stay in each place. Even so you'll have 2 or 3 days in each of the various destinations. You're covering a lot of territory and your pace is brisk but the trip is certainly doable!
I definitely think that eliminating Italy for now is the right move. You don't have time.
Since you are moving around so much I would try to pack just a carryon if you can. Hauling luggage around can be exhausting and you want to minimize this, especially for your mom.
I have done genealogical research in Europe and it is more time consuming than you might think. Finding the right office, language barriers, and bureaucracy can eat up a lot of time. Do your homework before you go. Know exactly which office or church you need to go to and create maps from your hotel, make a personal contact with someone who works there, and let them know ahead of time exactly what you are looking for. That way they can, hopefully, find what you are looking for before you arrive. Many places that deal with genealogy aren't open every day and you'll need to time your trip to fit the times that they are.
Can you be more specific about which parts of the Alps and Germany you will be visiting? Contributors here will be able to give you better advice if you can pinpoint your destination.