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You'll need at least 1100 EUR for that week, travelling on a very modest budget and eating food from the supermarket. To this come the cost of the flight ticket.
You won't be able to cover all of Europe in 7 days, at the outmost you can see one big city, another smaller one and one or two villages. If I was you I would not fly into Frankfurt, but into another city like Paris or Rome. I would also leave the kids at home. If you take them your whole trip would more or less revolve around the children and their needs- don't know whether this is what you want. One place that is really nice with young children is Paris, since you can go to EuroDisney. Germany can also be very good. Your initiary would really depend on when you are travelling and what you are interested in. I know a lot about travelling with children in Germany, feel free to email me.
Tip: I think at that age it is best to travel with a rental car or with a campervan. And best value for a family are usually not hotels but rental appartments or camping sites.
Source(s):
native German