They may authorize vaccines for 5-12 year olds any day, so if that occurs, that may address your soon to be 12 year old, and maybe even your 9 year old.
Keep in mind you are getting advice from a stranger on the internet, but the entry requirements at the moment are pretty well sumed up here: https://www.iatatravelcentre.com/world.php by clicking on Italy on the map.
Several key points: Your airline will be the one to make any determination as to what is allowable. In Italy, they likely will not check any covid related paperwork. You could call them, but I sometimes wonder if the random person on the phone knows any better than what you can find, maybe they have someone who can speak specific to covid requirements.
Italy does not exclude passengers from the US like other EU countries, so technically a CDC card is not required for entry, you will need it, or tests, once there to get into museums and restaurants.
So far as quarantine, if you look at clause 11, the last line indicates that children traveling with you, under 18, are exempt from quarantine.
So I believe, all you need is a negative test at time of boarding for each of you, your dPLF for each adult, and adults need to bring your CDC card.
The problem might be in Italy, if a museum, indoor venue, or restaurant interprets that you older son needs a CDC card or a test result, then you may be doing a test every few days on him as needed. They are easy to get, quick, but might cost you 22 euro a pop.
Hopefully they get testing approved for younger children soon, or maybe your Doctor can authorize a shot a bit early, not sure how much latitude they have.
As for your vaccination being more than 6 months old, that is not an issue. Italy does not specifically have an upper limit in place, and guidance on effectivity has been extended now to at least 9 months, probably soon or already to 12 months.