I don't see why it's not sensible to divide the diff types of geometry. If a lot of objects happen to span two child nodes to the root node (following a strict quadtree), then you'll see those objects all over the map. You could use loose boxes to avoid these splits, but then the terrain won't be as efficiently culled. Using separate trees will increase traversal time, but it will also make sure the types are handled correctly and in the best way possible. It's a matter of experimentation for your app, but for the general case, I'd go for two trees.

I think it should be feasible to store the geometry in the same tree, if the objects are all static meshes. If they're dynamic, use a different tree.