That sounds like a fun week coming up next summer! Cooking class, truffle hunting, wine tastings and Cinque Terre are going to make a wonderful and memorable trip with your friends. Since you want to include Cinque Terre, I would not suggest you go stay in the furthest possible corner in Tuscany so that you had to drive all of it to make your trip. Since you want to do wine tastings, I would suggest staying in Chianti and that is actually quite a wide area that gives you lots of choices as to where to stay, particularly areas that are not right smack in the middle of the wine region but more on the outskirts that will make it feel "off the beaten path".
I recently visited Osservatorio di Montignoso
and while it is actually quite central, it is at the top of a hill in an area that generally gets skirted around. Most traffic to San Gimignano, which is like 20 minutes away, comes from the east and south and this place is to the northwest. It feels off the beaten path and it was very quiet, the gardens were lovely and the pool area is very relaxing. Neither of the apartments sleeps up to 7 people so you could actually just rent the 2 apartments and have the entire villa for yourselves. Have the cooking class there (the owner offers them) and a Chianti wine country is just a short drive down the hill, as is the coast and Bolgheri, another top wine area. San Miniato is also not too distant, and the hills around there are perfect areas for truffles - although whether there will be any depends so much on the weather and we can't figure out yet where that is going.
Another super interesting place you could consider staying at, always central to the same area, are the Macinella Apartments
. These have views of San Gimignano and have lots of cool things and areas across the grounds, the owners are an interesting couple who love the place and keep adding to it. It would be perfect for your group, you could also add in some art classes if you wanted.
Keep in mind that for all the things you want to do, there are countless opportunities offered by small local businesses. As long as you don't head to the Antinori winery in Bargino (where everyone goes to), and instead go for local, small family-run or boutique wineries, you will get the off the beaten path vibe to every single thing you do. There are some awesome locals here who love sharing their passions and their love of the region, you will get to experience that in anything you end up choosing near to where you end up staying.
Let me know if I can offer more suggestions or ideas!