Our upbringing often determines our manners at the dinner table. In spite of that I have always feel that Christmas and New Years are special dinners and should be approached that way by all who sit down to enjoy them. I'm not being fussy but let us be fair to the person slaving over the hot stove in the kitchen.
My biggest wish is that everyone wait for the cook to sit down, then tuck into the lovely meal. I spent one christmas with a family and by the time the first people had been served out in the kitchen the last person to be served hadn't eaten yet. Not exactly family dinner in my mind. Think how that cook felt eating with maybe one or two and everyone else finished and up from the table.
As for them cleaning off their dishes, don't hold your breath. If they don't have the manners to wait for the hostess they sure as heck don't have the manners to clean up after them selves.
Another problem that guests and some family have is over serving themselves and sending a heap of perfectly good food into the compositor .No need of that and often that guest isn't asked back. In a case that I remember, it was my guest who took a big serving of sliced ham and then didn't eat it. Asked why, he replied that he "didn't like ham". Why take it, well it was the spirit of the moment or greed. That is yet to be determined. .
Thank the cook for the great meal even if it isn't great. She/he has done what they consider their best and they did invite you to join them in what can be a very private family gathering.Same goes for gifts you may receive.. Don't forget to give the big thank you!
Good luck with these gentle reminders.