Is "Kilenc fontos híd Budapesten van" incorrect? I assume that the different word order has a different emphasis, but are they both correct? Or does the English sentence imply a certain word order for the Hungarian sentence?
I would say yes and yes and yes.
Your version sounds like "Nine of the important bridges are in Budapest." Maybe they went there for a conference of bridges. Or we are taking an inventory of the important bridges of the world.