Tall order! So many places to go in the world! Do you intend to take each child to the same place, or each one to a different place? And how much money do you want to spend? All this will depend on the time of year that you can go, too, since the north and south hemispheres have opposite seasons, climate varies, and there are on and off seasons for things like huts and stores.
Here are some suggestions for week-long mountain trips around the world:
Tour du Mont Blanc, France/Switzerland/Italy. It's usually a nine day walk, but depending on how light you go and how far you walk each day, it is even possible in four. Ultra marathon runners do it in two. There are huts everywhere, all very well set up with good food and lots of people to meet. I met a father and his 13 year old daughter doing this and the Haute Route together.
The Pyrenees, as explained above.
The Picos de Europa, in northern Spain. Stunning and little known mountains that you will never forget.
Norway, again like above, for instance Jotunheimen, along with the northern areas of Sweden and Finland, like the Kungsleden and the Nordkalottleden.
The Lycian Way in Turkey.
The Larapinta Trail in Australia (but no huts... very wild).
The Overland Track in Tasmania.
Quite a few of the tracks in New Zealand.
The Japan Alps, especially the North Japan Alps, here where I live. Many very stocked huts, but can be very crowded. A lot of fun though... you will see Japanese at their very best up in the mountains.
The Everest Base Camp Trail in western Nepal, or the Annapurna Circuit (not one week though).
Climb of Mt. Kinabalu in Borneo.
Climb of Mt. Kenya (a little easier than Kilimanjaro)
The Drakensberg Mountains in South Africa
Crossing of the Atlas Mountains in Morocco
The West Coast Trail on Vancouver Island, Canada
Volcano walks in Guatemala
The Na Pali Trail in Hawai'i.
I don't know South America so well yet, but I hear the trails in Patagonia, in Argentina and Chile, are wonderful, like the Torres del Paine and Los Glaciares.