Winchester Model 70 .30-06. I don't see any need for a larger bore gun than the 06 given the game available in Sweden.
CZ 455 .22lr. this is a heck of a good rifle and for small game you need a .22. I would also look at a semi-auto if that's legal in Sweden. The Remington Speedmaster is worth looking at.
For varmints and medium game the .243win works great. Lots of rifles carry the .243. I like the Savage 12 VLP.
Most of the birds I've shot have been done with a Montgomery Ward 20ga. But that gun stays at home now that I have a Remington 870 12ga.
If you go for doves a semi-auto Benelli in 20ga or 28ga might be a good Idea.
A Winchester 94 in .30-30 as a brush gun. There are few rifles better suited to hunting in places where you can't see that far and you have to shoot quick.