I would not do an overall tour. The only place I would do an organized tour is inside Kruger if you don't feel up to planning it piecemeal.

I like the idea of 2 days in Maputo but may put it before Kruger if there is the possibility of flying nonstop from a Kruger airport such as NQP, to CPT which could streamline things a bit more. So LAX-DXB-JNB-MPM<2 days>Drive to Kruger?<4-5 days>NQP-CPT<5-6 days>CPT-DXB-LAX.

Since the exchange rate is very in favor of the USD, in Cape Town, I'd stay near the Waterfront, De Waterkant or Camps Bay areas renting a house or apartments. There are managed apartments which operate like hotels allowing for people to have their own space & privacy at reasonable price with services. For all but probably the 13 year old, one or two day trips to the wine areas is a good idea and the vineyard won't be boring to the 13 year old as there is plenty to see also depending on the vineyard. A trip down to the Cape to see the lighthouse, ride the funicular and see the penguins is a good 1/2-3/4 day trip. Cape Town and the area around it has a lot to offer from great food, to sites to see & a activities. We used Feel at Home in Cape Town & African Eagle and both were good. FAHCT for personalized tours and AE for longer distance group tours.

Be aware of the panhandling problem in cities but just be aware of yourself and you'll be fine.