Hi Amy
I did this trip last Christmas. We didn't have any snow and the sun shone but it was very cold in the evenings and on some of the trips in the Dolomites area where the high buildings seemed to keep the cold in. So you'll need your winter layers, a good coat, thermals and warm boots to combat the frost under foot. It was a lot warmer in Croatia and on our day trip to Slovenia there was a brief snow shower.
The dress code for dinner is smart/casual. People tended to dress up a bit for Christmas lunch and dinner. For New Year's Eve, everyone dressed up, some men in tuxedos and the ladies had their posh frocks.
Enjoy
Sylvia