Here are some fun games/activities from parties when I was younger.
Get 2 decks of cards and a few wrapped gifts. Hand out all the cards from one deck (you might want to make the deck smaller as there aren't so many kids). Then a parent holds the other deck and starts reading them out, as the children make a match they take a present from the middle of the circle. You continue untill all the gifts are taken and then the kids can start taking from each other. When the game is over the kids unwrap the presents.
A little bit before the kids go home put on a movie and then have a bag with many different colored popcorn bags inside, each attached to a ribbon/string. Have the string sticking out and let each child pull out a bag of popcorn. You could change this to bags of chips or some other treat.
Oh my gosh- the first game you described is a tradition in our family at holidays! We never had a name for it---always just called it "The Game"
as in "are we playing the game at Aunt Sarah's party next week?" Everyone knows what game we mean

When you play with kids do they have a hard time with the taking gifts from each other part? SIL always makes you feel terrible for taking her goodies! We play with unwrapped items though- I like the idea of wrapping them so you don't know what you've got until the game's over.