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Anyhow, during your 3 days in Asti you should see if you can fit some visits to the nearby area of the Langhe, Monferrato and Roero. It's a wine producing area famous for Barolo, Barbera and Asti Spumante. Alba is famous for truffles.
After that you have only 4 days and you are flying out of Milan. That means that the night before the flight back must be spent in the vicinity of Malpensa airport, especially considering that flights to North America will imply an early morning trip to the airport.
That means that you might have only 3 days. With 3 days only you should choose only one location. Since you are in Piemonte, and assuming the Langhe/Monferrato and Roero can be done during your first 3 days, I would concentrate in only one of the following:
1. Lake Maggiore and Lake Orta. This is also a good choice because it is a very short drive to the airport, therefore you could potentially spend even your last night in the area and drive to the airport in the morning.
2. Alps. Val D'Aosta and Piedmont region both offer great alpine experiences. You can drive, hike, take gondolas up to the top of Mont Blanc at altitudes of 4800mts (16,000ft), see the Matterhorn, Mt. Rose, Mt. Blanc.
3. If you prefer a coastal/beach ambiance, go to the Ligurian or even the French Riviera, spanning from Nice (once an Italian city, part of the Kingdom of Savoy) all the way to La Spezia (including, from East to West: Cinque Terre, Portofino, Laigueglia, Sanremo, Menton, Montecarlo, Nice and the Cote Azur. In 3 days I would choose the Cinque Terre in the East and maybe visit Portofino on the way there (or back). The car is not needed at the Cinque Terre, but it's up to you if prefer to keep it. It will stay parked most of the time at the hotel or the parking lot though. If you prefer the French side, Menton is nice and you can take a short drive to Montecarlo from there (Montecarlo is expensive to stay and parking is expensive).You can't fit Cinque Terre and Florence in 3 days. You must choose either one (no car needed in Florence either, it will stay parked the whole time).
I would choose no. 1 or 2 since you are upthere. If you want to fit a big city in the midst, nearby Turin or Milan will be a nice day trip.