Can I use a bike for commuting 10km everyday at Jönköping (Sweden)?
As long as the bike paths and roads you'll be traveling along are reasonably well cleared of deep snow, this would be no big issue.

You can order basic Schwalbe Winter studded tires cheaply (around 40euro a pair) from Bike24. You should probably be able to find a basic used bike for 110 euro.

Studded tires aren't strictly necessary for winter riding, but they do provide peace of mind on days with a freeze-thaw cycle.