Where are you flying from that you get to Rome at 22:30 Rome time? Most flights arrive in the morning, not at night. It sounds like you may be losing about 2 days to travel to Italy and may still have jet lag.
I would recommend that you take a taxi to your hotel, and crash rather than try to take a train to Florence first. Check with your hotel about food availability there or in the area that late. I suspect all of you will be hungry. If there is little late-night availability in the way of restaurants or markets near where you are staying, and you can grab something at the airport, do so.
So if I'm reading your itinerary correctly, you have 1 night on the plane, 4 nights in Rome, 2 nights in Florence and then 1 night back in Rome before you fly back home. Is that correct? I would definitely not substitute the Amalfi Coast for Florence. It's quite far and not that much warmer.
The average high in Rome in February is about 57 and low about 40. For Florence it is 53 and 36. For Sorrento the numbers are 53 and 44. Those numbers come from Weatherbase -- http://www.weatherbase.com/weather/country.php3?r=EUR&refer=®ionname=Europe. If you go there to take a look for yourself, click on the the F symbol at the top of the chart you get for the city of your choice. Each chart has a lot more information about rain, snow, etc.
I think you have too much on Day 3 and maybe not enough on Days 4 and 5 -- especially with a 10-year old. You will primarily be walking and standing in all the places you list for your sight seeing. Take a look at a map of Rome to see where these things are and you might be able to coordinate things a bit better.
My favorite maps are the Streetwise ones sold right here on the RS website and often in bookstores or at AAA. I like the Streetwise maps because they are laminated and easy to refer to. You can see the big picture as well as street and public transportation detail and locations of the major sights. There are also indices for the streets and things to see. Before you get a map, you could use Google Maps to get an overview of the city. Or this interactive map -- http://www.italyguides.it/us/roma/rome/rometravel/mapof_rome.php -- might be fun to explore.
My understanding is that the Trevi Fountain is still being worked on and will not be completed until fall 2015. That may be a big disappointment for you or it could be interesting to see what they are doing if you can get close enough.
Keep in mind that the sun will set in Rome between 17:30 and 18:00 in February. That's not necessarily a bad thing because many monuments will be beautifully lit after dark. But you may not be able to get into them very late. Be sure to check the open hours of the things you want to see. They will likely be shorter in February.
