Hi, Chani.
I was on the very first My Way Alpine tour in July 2013. As this was the year this tour was piloted, there were not as many dates to choose from. The July tour happened to fit my schedule that year, though as others have noted, September is a fine time in the Alps. I like to see the cows come down from the mountains. July 2013 was very warm, especially in Bolzano, Fussen, and Chamonix. I was in northern France this July and it was cool. The weather is a toss-up.
June vs. July: I would select earlier in June if possible for a very specific reason, which may or may not impact alpine tourism: the increase in folks traveling as summer school holidays begin, including what appears to be a growing number of organized active teen tours.
I started my pre-tour in Innsbruck and took the train to Salzburg. Alsace after Chamonix sounds like fun.
Mobility: I agree with what Sara said. There are many options to strenuous hiking. Some of our tour members were definitely there for the more active hiking, while others of us, while steadily on the go, opted for a more relaxed tempo. I did notice, somewhat to my chagrin, that it took me about twice as long to complete any walk compared to what was posted in the tour book or on local signposts.
In sum, given my druthers, I think I would select June over July, but the July trip I did was very enjoyable, offered lots of opportunity to be with others or be on one's own; opt for off-the-beaten path experiences or go with the flow - even Hohenschwangau with its many visitors was charming. My biggest wow was overcoming my fear of steep mountain cable trips. Fantastic!
I don't think you can go wrong for any of the dates for this trip. Check out the 2013 scrapbooks here: https://www.ricksteves.com/tours/scrapbooks/tour-alum-scrapbooks.
Debbie