I would limit the number of moves, especially with the two boys along. Pick four cities and spend a week in each one. This allows you and the boys to get acquainted with a neighborhood and places to shop and eat. Take side trips by train as weather permits. Move from north to south, maybe one week in London. one in Paris, one in Switzerland, and one in Rome.

Rent apartments so you will have separate bedrooms, a living room and a kitchenette so you can fix simple meals when you tire of eating out, and have snacks and drinks available during the days. It will be no more expensive than a hotel, and will have a clothes washer, too.

I assume you are prepared to have bad weather on the trip, and can find ways to entertain yourselves when it happens. (Remember the TV probably will not be in your lamguage, so that is out.) You could plan on the Switzerland part of the trip for snow fun.

Yes on reducing the number of stops and staying a week in each

Yes on renting apartments

The weather may be awful or it may be brilliant or it may be a combination. What it is certain to be is dark early. You don't say where you are coming from (guessing Commonwealth because you wrote "whilst") but Rome is as far north as New York, and while touring cities at dusk and in the early evening is wonderful, touring the countryside is not.

Every one of the many flats we have rented has had television with at least one or two English language channels, but wifi is even better if you need to amuse the youth and keep yourselves informed.