Ciao Weilyn,
Your plan sounds perfect! It is actually what I highly recommend, using one base in Tuscany and doing day trips.... Florence is pretty central and Tuscany isn't that large - although being pregnant, play it by day and see whether you want to alternate between a day out further away from your base and another in the immediate area, there is plenty to see and do almost everywhere.
With a car, I recommended staying outside of Florence completely. You don't need a car if you're in Florence and would actually have to find a place to park it while you don't use it at all. It is better to be outside of Florence and do a day trip into Florence in that case.
So here is what I suggest as far as places to stay that offer cooking classes and are close to Florence:
Villa il Poggiale
Poggio di Gaville Villa
La Medicea villa
Torre a Cona
Residence Santa Maria
Podere Casanova doesn't offer cooking classes directly but would be great for kids, particularly for the pizza night if your kids are old enough ;-)
As far as itinerary, definitely do day trips to Florence, Siena, Pisa,
San Gimignano, the
Chianti area. The castle of
Monteriggioni near Siena is really neat as well.
June 6-13 is really soon, I hope you can find availability! The links above will take you to the official sites so that you can contact the owners directly and get personally in touch and express your needs and desires... get moving!

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