Piedmont is landlocked, it doesn't have any coast and it is definitely not South Italy.
The city on the French border would be Ventimiglia, but it is like Calais in France, where those attempting to cross the border illegally will gather. They get to Italy, not the mainland, the islands, through boats, and then make they way up and Ventimiglia is struggling.
The Italy you have in your mind, you won't find it in Ventimiglia, or even Milan. Milan is a metropole, it is big, fast paced city, with long working hours and traffic, great shops and amazing aperitivo. Then you have the opposite, those dying villages that make the news, and the 1$ scheme. This is not an option for a young family, not because you can't access it, but because you will hate it. You might as well throw your kids on a rock in the middle of the sea. There is nothing. No school, no shops potentially, you are in the middle of nowhere on a hill , not even goats roam around. After the kids are gone, yeah, you can try this adventure, but now, forget about it.
Italy as a country is not very old. It was unified in 1861. This means every region is very different to each other, and you shouldn't throw it all in one bag. Piedmont and Puglia have little in common, from the food to landscape. I would pick Puglia over Piedmont. I would pick the coast, especially keeping the school holidays in mind.
One thing many don't know about Italy is that you have more than three months Summer holidays. School will finish between 4-10 June and start again after the 15 September. And you have one more year of schooling. Kids do their "maturità " at the age of 19.
Your questions about culture shows how little you know about Italy. There is plenty of culture. Even very small cities have a theatre, with seasonal tickets, and in the Summer, you have opera in the main square or inside monuments, not just Verona, many places, such as the Castello Sforzesco in Milan and for very little money you can enjoy it. In small towns, it will be free. Italy has more art than a lifetime will allow you to admire. From churches to museums, there is plenty anywhere you look. Then so many cities have a " sagra", and the streets will be closed, filled with tables and everyone sits together enjoying whatever food they are celebrating.
Accomodation will vary. We did a short posting in a main town, and our flat was in a building from the Middle Ages. Heating was centralised, meaning the whole building shared a big furnace and there were local laws when it was turned on (dates and hours per day). We rented a flat in a seaside town and it was modern. Overall , quality is far greater than Sydney where we are now.