I am guessing you are traveling by train as the distances will take too long to drive.
A Swiss family pass will allow your 12 year old to travel for free. I have only been to the Junfraujoch area about 5 times, once with elderly parents, once with a Girl Scout troop of 14 year olds, and other trips with my family. My first impression of your plan is that it is so much traveling and moving every night to a new hotel might wear you out.
Is there a possibility of concentrating on the Bernese Oberland area as there would be plenty to do in the area between Interlaken and Jungfrau. Interlaken is a city on the lake, and there are lots of villages and activities in between the city and the mountaintop. The train trip up to Jungfraujoch is discounted (with a some rail passes), if you take the first train up. You can catch the train from many villages along the route--the farther up you start, the cheaper and shorter the trip up to the top .
You can stay in the valley--in Lauterbrunnen--or part way up the mountain in a car free village like Wengen.
I love Wengen. Wengen offers shops, a Co-op grocery, restaurants, a community center that has programs about the culture of the area, lots of rental properties for short stays and hotels and is accessed by train. From Wengen there is the cable car to Mannlich, then down to Gimmelwald. Beautiful views!
There is the Bernese Oberland Area train pass or the Jungfrau Travel Pass which allows free/discounted travel on trains, cable car trips, discount to Jungfraujoch, buses, etc. They have a three day pass and you could plan your time to make maximum use. Jungfrau .ch is the site. For example--the 5 day pass is also good for free tickets on the boats you can catch in Interlaken for a cruise around the lake and perhaps some stops to see other towns. Trummelbach Falls can be accessed from Lauterbranen by taking the bus a very short ride and very easy to locate. The falls are inside the mountain!
Take the cable and funicular up to Murren--opposite side of the valley from Wengen. Murren is another car free village. From Murren it is a flat level short walk to the cable car that takes you to Schilthorn--rotating restaurant from 007 movie. Could be foggy or a beautiful view depending on the snow cover and temperature. Have a meal, and walk or take cable to village of Gimmelwald--Heidi like area--beautiful views, ability to purchase eggs and cheese from the residents to cook when you get back to your chalet. Another no -car, very small village.
Milan is a big city; the church and the Last Supper are definitely worth seeing, but I think if you focus on covering less ground, you will have a more relaxing trip and still be able to see a lot. Trying to find your way around a big city when you only have a day is difficult. If you are traveling by train, then you are talking about taking your luggage on and off the train, getting to a hotel to drop the luggage off so you can sight see, trying to get oriented in the city you are in, then packing up and going to the next town. Imagine pulling into a big US city on the train, having to load your luggage into a taxi (unless you are backpacking), head to a hotel to drop it off, then figure out what you can see in the few hours you have, crash, catch another train in the am....I get tired just thinking about it.
My advice would be to pick 3 places to "park" and take day trips from this base. You can fly into Milan and head to Lugano immediately--skip the city of Milan this trip. And then another base in Bernese Oberland area, and then one in a city where you can explore in more depth for a couple of nights.