It seems that the usual thing to do is put the wormhole in orbit around your planet or whatever. That's based more on science fiction than physics, albeit scifi by smart knowledgeable people.

I'm not sure what to think about gravity "reaching through". Gravity and the wormhole are both just curved spacetime. Maybe the larger body distorts the wormhole throat in a way that looks like "reaching through"? Certainly, you need to have some effect that causes energy conservation when you make a perpetual water wheel by suspending one mouth over another and running water through it...