New Year's Eve is the only event celebrated by the entire world.
New Year's Day shows the influence of: Christianity; the Church; and the state.
"A.D." always goes before the year. It stands for Anno Domini, which is Latin for "in the year of our Lord.
I don't go in for "common era," because I am a Christian, because everyone knows the dating is from the birth of Christ, so why try to pretend otherwise, and because while I assume most people who opt for "bce/ce" do so without any animus toward Christianity, some do opt for this language as a way to stick it to Christianity.
I haven't taken part in a New Year's Eve celebration for a few years; I'm not against it, but it doesn't do much for me. So I took the early Mass this morning, allowing the other priests to sleep in. I was still up till almost 1 am; even had I wanted to sleep, the neighbors were out in the street around midnight, banging pots and pans, and shooting off fireworks.
How do other nations celebrate New Year's Day? Too bad they don't have (U.S.) football, but I'm sure they don't feel terribly deprived.
In the old calendar of the Roman Rite, this day was the Circumcision of the Lord; so was there another day for the Theotokos?