Where I come from everyone says ‘schoolbag’, but all my students say ‘bookbag’. Is ‘bookbag’ just a chinglishism derived from ‘shubao’, or am I just hopelessly ignorant? It’s almost certainly the latter, but I need to know!
Nothing wrong with it.
At least, the books will fit into the book bag.
The school will not fit into the bag. 
Good point. Makes sense. I was ignorant, but you have shown me the light. 
Um, we all say that in the US. At least everyone I know. “Backpack” is also common but obviously only when it actually is a backpack. But I’ve heard people call a backpack a bookbag, if it was used for that.