You've gotten lots of good advice and acraven is always a treasure trove of insights. I strongly recommend their advice is followed as far as booking in particular. The two items that stand out for your destinations are the Coliseum and the Anne Frank House. If those are priorities, you need to check the release schedule and set an alarm on your phone or you're likely to miss the opportunity. If cost is no object, you can book a third party guided tour of the Coliseum and those will be available much closer to your arrival date. Not only is it wise to book hotels with free cancellation, but tickets and tours through viator, tiqets, etc, will offer free cancellation up to 24 hours before the event on many things.
As a budget traveler, I avoid extra expenses, but my partner has reminded me often that I am paying for my time there already with hotels, tickets and time off work. Paying $15 more on a skip-the-line to gain an hour Rome instead of spending it in a line is an excellent value. (Some skip the line tickets don't even cost more... they just need to be booked ahead.). Taking a taxi from the airport in Paris is a good value. Less so with the others. (Express trains/trams are offered specifically for airport transport.). If easyjet is going into CIA instead of FCO, get a taxi.
We found easyjet to be perfectly suitable. They won't give you so much as a sip of water for free and if you have much luggage, it will probably cost more to bring than it costs to bring yourselves.
I am the opposite of acraven in terms of jet lag so I have to root for direct flight on the return. I find if I adopt the new schedule and go to bed early my first night in Europe, I'm generally good to go, while it takes me 4+ days when I return. Things are exhilarating on the way there, but delays and connections are painful on the return when you are looking forward to just being home. That said, Rome is hot. and crowded. (It's my favorite city in the world at the moment, so no aspersions, just a fair warning.) If that is an issue, the later in the year you go, the better.
I personally enjoy my first day and make use of it since most transatlantic flights arrive around 8am, but the first day in each city will be at least a little eaten up with getting your bearings, checking into your place and, unless you use a taxi or walk, acclimating to a new public transport system. It also depends on how well you sleep on a plane and if you get by well on a few hours or less. If you typically stay out late, the first day will feel abbreviated as it would be miserable and unwise to do so. I am up til midnight at home and I crashed around 8pm on each overseas trip. It makes sense for a lot of people to get checked in, take a stroll, get some food, rest and sleep on the first day. I don't have the temperament to be in my hotel when a city I want to see is just outside and I don't have the ability to nap, so take my thoughts with a grain of salt.
I think your itinerary is delightful. I've been to all of those places and I think you've paced it really well and have given appropriate time to each place. I wouldn't bother with a day trip from Rome with only 4 days, but Paris much (much) smaller. I find one or two "big" things a day is ideal. Room left for wandering is the difference between visiting a city and loving it.
I am a bit of a maximizer, so I've ended up flying between cities quite a bit while in Europe as it's far less expensive than returning from the US. I strongly(!) prefer train travel, but it's been worth it when I've flown if it makes the itinerary make sense. It does have a way of wearing down one's patience and energy more than train travel. Busses are about as enjoyable as an airplane but at least they don't have the hours long commute/security/boarding.