The Hamlet Project

1) Claudius was in the wars, too? We know about Hamlet Senior and his sledded pole-axe. But Claudius was in battle?
2) Against the French? When did the French and the Danish get into military scuffles? This is not current, obviously, since Laertes was off to France at the top of the play.
3) But at one point, the Danes were fighting the French on horseback. I guess each nation has its own specialty. The French are good on horses? This seems weird. How did THAT come to be? Or is it just something Shakespeare made up? How about some nitty gritty literary military history? I’d like to know where all this is coming from.