Can road disc calipers be mounted on a mountain bike? Are the mounting posts and position for road disc brake calipers the same as or similar enough to mountain bike disc calipers that you could put a set of road disc brakes on a mountain bike or vice-versa? 

I'm aware that this generally requires road brake levers, and that you also need to make sure any drop bar levers are compatible with your derailleurs. However, my question here is narrowly about how you would convert the brakes of a disc-equipped mountain bike to something drop bar compatible. 

If that can be done, what are the downsides to be aware of?
Are the mounting posts and position for road disc brake calipers the
  same as or similar enough to mountain bike disc calipers that you
  could put a set of road disc brakes on a mountain bike or vice-versa?


IS and post mount calipers are cross-compatible and can be seen on both MTBs and road bikes. Flat mount clipers, on the other hand, are a new thing that was designed specifically for road-type usage, with very limited presence on MTB frames (some small builders and Cannondale bikes). You can put a post mount caliper on a flat mount frame/fork using an adapter, similar to IS-to-post-mount.


  However, my question here is narrowly about how you would convert the
  brakes of a disc-equipped mountain bike to something drop bar
  compatible.


Certainly possible, but details depend on what calipers MTB has. Shimano hydraulic calipers are compatible between road and MTB levers. In fact, BR-R785 was an XT. I don't see why you can't just get a set of any Shimano hydraulic road levers and put them on. No idea about SRAM, though. You should also be aware that mechanical MTB/road calipers might have different leverage ratio, like BB7 does.


  If that can be done, what are the downsides to be aware of?


Except for existing parts compatibility I see no reason why anything might cause trouble. I'd be concerned about geometry though, drop bar will increase the reach significally, you might want a shorter stem.