Huh...well I lived in Germany for 5 years, but I don't know that I'd be able to answer your questions with regards to the US either!
How could an athletic shoe and apparel store be successful in that country?
Offer cost-conscious, but well-made/reliable products. If there was an Aldi Mart for shoes, I bet it would move volumes...
Should it rely on style, function, or both?
Both. I've found that Germans are very style-conscious, but definitely are big on finding the right function for a shoe. I mean, this is the land of the "zu Hause" shoe (aka Birkenstocks).
Which style or sports focus would be most popular?
I expect this would be determined by region - southerners have their hiking thing, football's popular throughout, etc.
Have you noticed any trends in casual shoes and apparel in that country?
That they're all sturdy/well-made and well cared for. Sneakers were decidedly less "in", unless it was a specific style with the dance crowd and the like. I remember getting shit for matching Nikes (or NBs?) with jeans one time...