What do we know about it? Did they have battlemages? What kind of magic did they use? Anything. I'm making a nord battlemage in Skyrim and I wonder what to do with him.

Traditionally, Nords and Magic don't really get along. There's some Shamanism present in the culture, and Priestly Magic would not be snubbed, but for the most part Magic is not something encouraged to study for its own sake.

It boils down to the Nords not having a really good history with Magic. You've got the Falmer - who were just as, if not more magically inclined then the Altmer, Queen Potema didn't do Mages any favors with her behavior, and it's still generally accepted (even though it's patently false) that the College was responsible for the Great Collapse of Winterhold.

Going by what the guards say, Restoration is seen as a worthy school of study. They frown upon Destruction and Conjuration, however.

Huh? Older nords and dragon priest were good/great magicians, the hate for magic did not existed, though they may preferred the power of thuum.