the Hallsatt photos were somewhat romanticed for publicity. The reality is in December they are unlikely to have quite as much snow (the flights for ski trips don't start until mid December and they are using higher resorts) and even less likely to have bright sun and blue sky. Because of short days and high mountains around, in mid December the town it is unlikely to see the sun before 9AM and it may be going dark by 3PM. Even the villages with less mountain shade will be lucky to get much more sun - if it is out. If cloudy it may never really get light.
Innsbruck and Bressanone/Brixen are close enough to each other to do as a day trip in either direction but the sun works the same there, although as you head south there's more chance of getting some. "Dolomites" is a huge area but assuming you mean the west side of the Italian section it would be very hit and miss. Go too high and everything will be gearing up for ski season OR closed and abandoned until spring (which includes many roads). Further south you may have more success.
Consider Innsbruck - Bozen - Lake Garda as a natural tour, Hallstatt is way off in the wings. Innsbruck Christmas market will be going strong all December https://www.innsbruck.info/en/innsbruck-city/events/advent-and-christmas/christmas-market.html
Bozen is a little further south than Brixen and is the gateway to the Vinschgau are where you may find some hiking. It also has buses eastwards into the more well known Dololmites as road conditions permit. And it has a christmas market as well
http://www.bolzano-bozen.it/en/bolzano.htm
http://www.christmas-markets.it/en/christmas-market-bolzano-south-tyrol.aspx
http://www.vinschgau.net/en/welcome.html
Lake Garda is another hop southwards and should be brighter. http://www.visitgarda.com/en/garda_lake/