There are two parts to your question: How do you get the younger kids to participate and how do you keep the older ones interested.
Can the older ones help out by previewing parsha sheets with the younger ones (those sheets are NOT a test!) and giving the kids rides by the zemiros? Some kids like to.
Definitely give younger kids break time. They can go play and you’ll call them back. You can give them break time between each course!
R. Pesach Krohn says that in order to have a successful Shabbos meal the father needs prep time. What is his Dvar Torah? What is his story (we can start a new thread for book recommendations)? Does he have any hashkafa issues he wants to discuss? Does he have a sefer that he learns a little from every meal (we can start a new thread for recommendations)?
I would start a behavior system for the fighting. It can be special Shabbos treat for good behavior, or a check after Shabbos which can be used to buy a prize.
Good luck! Before you know it the kids will be bigger and you’ll wonder where all the noise is!