Can I use a bike for commuting 10km everyday at Jönköping (Sweden)? I'm a French student going to Sweden for one year and I would like to know if I could use a bike to go to school even in winter for less than 150€ (or 1500 SEK).  

Is it possible without having a bad time when riding and or will I fall every five seconds due to icy roads?  

If it's possible I would like to know which type of bike should I use and if I have to add accessories like studded tires or tire chains.

Here's an average weather of the city:
https://weatherspark.com/y/78217/Average-Weather-in-J%C3%B6nk%C3%B6ping-Sweden

And some details about my route:  

Start altitude:                  102 metres
End altitude:                    111 metres
Maximum altitude:                129 metres
Minimum altitude:                93 metres
Distance:                        5.1 km
Total ascent:                    54 metres
Total descent:                   45 metres
Maximum gradient ascending:      5% at 1.7 km
Maximum gradient descending:     6% at 0.6 km
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.