We took one of our two sons to Italy when he was just turning 9 --- he slept on the plane and didn't seem to be bothered at all. His older brother, who was 16, joined us in Italy after first spending a couple weeks in China with his youth orchestra, flying home to Minnesota for half a day, then getting on the plane to Italy the very next day --- again, it just didn't seem to affect him.
Our two grandchildren going to Italy with us were an entirely different story at ages almost 4 and a baby of 5 months. They slept on the planes, were great little travelers, very cheerful, and then just could not sleep the night of the day we all arrived (early that morning) --- they kept their parents up off and on all night. Then were fine from then on.
So, yeah, different kids are different. We are planning to take our grandchildren to Italy again later this year, at ages 9 and 6, so we'll see how that goes. But we and their parents will for sure talk with them about jet lag and what to do if they are awake at night and will provide them with quiet entertainment. Books and flashlights, iPads and headphones, coloring books, stickers, etc. Ideally, new stuff. Plus they will hear some good advice about what is worth waking up your parents for and what is not. It will be a good idea to separate our grandkids in different rooms since at our cabin and when doing a sleepover weekend with us they are inclined to wake up super early and start horsing around together. Maybe one parent with each kid.
We always stay in apartments when traveling with kids --- I would think sharing one hotel room would be gruesome with your three kids, no matter how reasonable they are. I would also think it would be good to stay the first few days in the same place, just to allow everybody to feel settled (and remember where the bathroom is....)
Many people disagree, but we think it's totally, totally worth it to take kids on such trips, even if the first night is rocky. Good luck with your adventure!