What time of year? I'm 67 years old and a basically healthy, though not athletic, traveler. I typically walk from 6 to 8 miles a day and occasionally hit 12 or 13. For me 2 or 3 nights would be a short stay. I try very hard to have at least 4 nights somewhere. I do that by picking bases that offer both sights of their own and (unless we're talking about Paris, London, etc.) opportunities for day-trips. I just find the business of checking in and out of hotels and rolling my suitcase to and from train stations a drag. That doesn't mean your mother will feel the same way, but we are about the same age and may prefer to settle in a bit longer than you are proposing. Also, she probably doesn't walk as much as I do at home; I do not own a car.
I'm always a bit concerned when I see an itinerary listed like yours is, with the same day attributed to both the departure and arrival cities. At a glance, that tends to suggest you have a lot more time at each location than is really the case. I'd rather see it listed like this:
Day 1: Arrive Vienna (2 nights). Palaces, horses, treasury. [Is Day 1 the day you fly in? That's not a full day.]
Day 3: Salzburg (3 nights). Possible day-trips to Zell am See and Innsbruck.
Day 6: Luxembourg (2 nights). Battle of the Bulge Cemetery, castles, hiking.
Day 8: La Rochelle/Ile de Re (4 nights). Or possibly stopping in Strasbourg for a night or two.
Day 12: Somewhere (2 nights).
Day 14: Fly home. [Or do you fly home on Day 15?]
So now we can see that you have just one full, relatively non-jetlagged, day in Vienna, plus a few hours on your arrival day when you may be zombified. As I've gotten older, it has become even more difficult for me to sleep on the overnight transatlantic flight, and jetlag hits me harder. Maybe your mother will be luckier.
Large cities like Vienna are harder to get oriented to, and you're allowing very, very little time there. I don't think you have enough time for what you want to see. Then, less than 48 hours after your arrival you're going to packing up and heading to Salzburg.
In Salzburg you have two full days plus some hours on Day 3. That would be generous just for Salzburg, but you're considering two day-trips, which seems as if it leaves little time to relax in Salzburg. But I've never been to Zell am See and am not knowledgeable about traveling there from Salzburg. Still, two day trips would seem to make for an awfully short visit, given that it's your mother's birthplace.
Luxembourg City has some historic architecture but isn't usually part of a first- or second-trip itinerary. Do you have a family reason to visit the Battle of the Bulge Cemetery? Do you plan to rent a car here? Castles (plural) and hiking sound as if public transportation could be a challenge. And you have just one full day here plus a few hours. Too much on your plate, I think.
Strasbourg has a beautiful historic area, and nearby Colmar and the little wine villages are popular. I think most people would find them more engaging than Luxembourg, though I don't know about the castles around Luxembourg and am not much interested in that sort of thing anyway. A short stop in Strasbourg would probably not give you time to see anything except Strasbourg. I think more nights would be advisable so you could make Alsace a real destination.
I'm running out of room so I'll make a separate post with comments about La Rochelle and Ile de Re.