Oh yes, everything remains open, maybe wineries will be closed on the strict main holidays (December 25th and 26th and Jan.1) but restaurants will be open for sure even on those days.
Museums tend to rotate... some will be open for the occasion, some closed, depends on the museum!
Will you have a car to move around? Despite it being the start of winter, I'd still recommend you take a look a this itinerary for ideas of places to see and visit:
http://www.discovertuscany.com/itine...n-tuscany.html
With such small kids, you'll have to remain flexible and adapt to their needs but it is a good start as to what to see and do ;-). Cortona is beautiful so it is a great place to be based in, particularly to see the south of Tuscany.
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