“I was thinking 11 nights total (overnight flight for 2 days coming and going) and then 3 nights in each city traveling to next on day 4. For example, if we arrive in London on a Monday morning we would not leave for Paris until Thursday morning giving us 3 full days in London.”
I plot out my trip based on where I will be sleeping. Based on your info, you really only have 9 nights “on the ground”:
Night 1 arrive London: check in, light sightseeing
Night 2 London: sightseeing
Night 3 London: sightseeing
Night 4 Paris: travel from London half day, half day sightseeing
Night 5 Paris: sightseeing
Night 6 Paris: sightseeing
Night 8 Rome: flight from Paris half day
Night 9 Rome: sightseeing
Hopefully this shows how little time you truly have in each place. Traveling from Paris to Rome could take a big chunk of the day, considering how long it takes to get to the airport, how long you have to be there before the actual flight, actual flight times, how long it takes to get from Rome’s airport to the city. This could easily be 6-7 hours.
That said, I will echo the recommendation to pick two of the cities and spend more time in those two places. If Rome is one of your choices, start there before it gets too hot. Then fly home from the second city.