I let DD open any gifts she might get from visiting Santa in the mall and if people visit on Christmas Eve then she can open those gifts.
Everything from me is wrapped and put under the tree once she is in bed on Christmas Eve, so she wakes up to find lots of presents because Santa has been!
Santa comes to our house on the Winter Solstice and he only fills stockings. Everything else is from us. Our kids get three gifts: something they want, something they need, and something we want them to have. All of our gifts get put under the tree as they get wrapped. I always try to have at least one gift ready to put under the tree the same night we get it decorated. I love watching the tree get filled up with gifts as the days & weeks go by. :p If someone is visiting us, or we're visiting them, and gives a gift, it can be opened right then & there so the giver can see their reaction & get thanked in person. Anything that arrives in the mail is put under the tree.
When I was a kid my dad usually worked both of his jobs on Christmas Day because no one else wanted to be there & he liked the double or triple overtime pay. We'd spend Christmas Eve with his side of the family, then he'd take us to our mom's church in time for Midnight Mass. We'd move all our gifts from his car to hers, then he'd leave & we'd go home after the service with her & have Christmas Day with her side of the family.