We have spent significant time in Venezia in December and also spent last Christmas in Ortisei.
First, as to closures. It's pretty quiet Christmas Day wherever you go, HOWEVER, a lot of people are traveling and restaurants, etc, are open. You just will not find museums operating. Churches will be open but of course mass will be held. Christmas markets in some places extend past Christmas Day, in particular Bolzano and Ortisei markets go to New Years.
I love wandering around Venezia in pre-Christmas weather. Tourists are few, the lights are lovely, but I would probably spend the actual holidays (24-26) in Ortisei. The mountain resorts are really set up for activity even on Christmas Day.
Ortisei versus Bolzano: no comparison. Ortisei is actually in the mountains and an adorable alpine village. There is opportunity to hike or ski on both sides of the valley. You buy a pass with points on it to use on the lifts. Last year we hiked at Rasciesa on Christmas Eve Day, and on Christmas Day hiked across the Alpe di Siusi from Mont Seuc to Compatsch. Lunch was available at rifugi and fabulous dinners served at our half-pension hotel. We did NOT have a car, by the way. We hardly ever rent a car and yes, winter driving can be hazardous. I have had friends with an SUV and snow tires turned around by the Carabinieri and not allowed to go down from a ski resort. Threw off their entire schedule. The buses were allowed to pass. I was in Innsbruck many years ago, and I'd say Ortisei has it beat hands down. The Val Gardena has a very active association for tourist activities called Val Gardena Active. For non-skiers, they offer guided hikes, snow shoeing, sledding , etc., almost every day.
Bolzano: easy to spend 1/2 day on the way to/from the Val Gardena where Ortisei is. See the market, Otzi.
Depending on your preferences, a half-pension resort hotel might be a nice treat over the 24th-26th (Christmas Eve through Santo Stefano) in a resort location like Ortisei. We did this last year and it was delightful to have dinners and breakfasts provided, a nice pool and spa, shuttle service to the lifts.
You'll find lots of speck (a type of prosciutto particular to the north, canaderli (dumplings, also northern), and all over Italy panettone a Christmas bread. In the Val Gardena weinerschnitzel is side-by-side with spaghetti al pomodoro on menus. It's an interesting fusion due to the Austrian heritage.