It seems like an awful lot of activity on (what could be) really hot days. But, that said, you know your friends and the boys, and I do not.
If you live in a location that might have summer temperatures similar to Rome, maybe give it a "trial run" -- obviously different activities...but something very active outside 11-1, then go for a bike ride 3-7 and see how everyone does during the sort-of simulation. I can make my predictions, but they would be based on very little knowledge. And, hopefully this will not be on the first day of arrival in Europe (jet lag).
Pompeii is fabulous, but it might also be trying (baking in the mid-day sun). I would suggest an early-morning tour of Pompeii for your group, having the guide show you a single example of the key architectural/cultural features. Hats/sunblock appropriate. There is really no way to see everything Pompeii has to offer in one day....we met some people who did it over 2-3 days (which would be an overdose for me). But, if you all may never get back to the area around Rome, definitely squeeze in Pompeii.........interesting and thought-provoking, and the boys will get a few giggles, too.
Lucky kids to have such a fun, educational trip, and I hope they truly appreciate what you kind parents are doing :) Enjoy your trip. Let us know how it goes. I have a feeling the gladiator school will be a "keeper" on the itinerary......what young boy would not want to experience that?!!?!