I don't know about buying a bicycle in Europe, it's going to be more than you'd pay for an equivalent bike in the states. Deal, or not, you're still going to pay high VAT. It's 20% here in Italy.
That being said, I live in Northern Italy, a bit over an hour north of Venice. Cycling is great here, but I agree that the larger cities can be pretty hectic, and the drivers are scary.
1 week per location isn't that long to see sights in Paris, Italy and the like. You'll be busy, even without a cycle.