You have 4 locations that are too far from each other. You'll use up 1/2 day every time you move. It's more than 3.5 hours on the train frrom Venice to Rome. Figure at least 1/2 hour to get to the station and another 1/2 hour from the station to you hotel. That's 5 hours gone, not including packing/unpacking, checking in/out, and traffic delays. And that's your shortest trip. All the others require flying, so more time to get to/from airports plus all the extra time at the airport to check in, go through security, boarding and disboarding time, and baggage collection. Flights within Europe, even on legacy airlines, often have lower allowances for cabin baggage, so even if you aren't checking bags on the transatlantic flight, you may have to check your stuff for the other flights.
If you are flying RT Paris, there's the added wrinkle of a "connection" in Paris to Venice because it sounds like you plan to buy separate budget flight to Venice. Since the flights are ticketed separately, chances are you'll have to go through passport control and baggage collection in Paris, then check in for your next flight. If there's any delay and you miss your European flight, you have no recourse.
3 nights in a city gives you 2 days of sightseeing, 4N = 3 days, etc. Your first full day in Europe is likely to be ineffective while you are getting over jetlag. OTOH Venice is the best place to relax and soak up the atmosphere.