Sweden in Jan from Skavsta
This one is tough, very tough (based on 8 years experience living in Uppland, N of Stockholm).
First, the bad news: January is possibly the worst month for a visit. It is dark (perhaps 6 hrs daylight, if the sun is out), it is cold (-20C not unusual), and it can snow, hard. There is likely to be snow on the ground, which slows driving, blocks unploughed forest roads (unless you have a 4x4), and makes walking in the forest difficult. Birds can be hard to find, except at feeding stations and any open water. Sometimes, invasion species arrive and add interest: pine grosbeaks, crossbills, hawk owls, Siberian tit, but not this winter (so far). Even the waxwings have gone. Owls and woodpeckers have not started calling in Jan.
Then, better news: Sweden has an excellent system for reporting birds, called 'Svalan' (the swallow), so you can see, day by day, whether anything of interest has turned up. There is even an abbreviated version for mobiles. Go to:
http://www.artportalen.se/birds/todays.asp
and navigate to days and regions of interest. Click on 'Today's bird for WAP' to get the URL for mobile use of any page of interest. There is also a search system (-> Start -> Search records), which works with English names.
As for particular sites ... Skavsta is a bit far south for most of the species requested. If you can get up to Uppsala, things are better. A few km W of Uppsala is Fiby urskog, which is a nature reserve with hazel hen, three-toed woodpecker, and more. It probably offers the best chance of seeing these quickly. Just walk around the paths ... hazel hen is really difficult to find: consider yourself very fortunate if you find it in one weekend! For the woodpecker, just listen for it gently hacking away. Black woodpecker is there (listen for the calls). Grey-headed is more difficult: best chance (in Jan) is if one is turning up at a feeding station regularly (use the website above to find out about that), and be prepared to wait, possibly hours, in cold weather.
Owls: tough in Jan. Best bet is to drive along roads in the forested areas around Uppsala, especially to the N, and check on the fences. This can give Ural owl, even in the day. No hawk owls at the moment, nor great grey. Tengmalm's is always exceptionally hard to see, except at the nest. Pygmy can be seen at dusk, often on treetops, or calling. Again, drive forest areas.
All in all, I would recommend Fiby as the best chance for something. It will take a whole day. Be prepared for a lot of driving.
All a bit negative, I am afraid ... but I hope it helps.