Starting with the background provided in this question, how would a dragon outfit himself/herself (avionics, power systems, etal) for operating under the Instrument Flight Rules? How would the process of obtaining their instrument rating work? Also, how would this differ if dragons were susceptible to spatial disorientation vs. if they were resistant or immune to it? What would a dragon sound like in ATC's ears? How would they file flight plans? What weather minima would they have?
Remember that the air is the dragon's natural environment, so dragons will probably be much harder to disorient. Disorientation is the result of environmental cues (rotation rates and accelerations) in a maneuvering aircraft that differ greatly from what humans experience on the ground. Our brains never evolved to handle rotation in multiple axes, and fail to correctly figure out what's going on. A dragon (or any flying animal) has a brain evolved to process flying, and so will be as resistant to disorientation as a human walking along the ground.
Dragons will probably have a different definition of IFR flight than fixed-wing aircraft. Since they fly slower and are vastly more maneuverable (due to highly variable wing geometry), the risk of collision is reduced even in low-visibility conditions. Flying through thick clouds or heavy rain will still carry the risk of collision, but I doubt dragons would do much flying through thunderstorms; after all, they aren't protected from the elements by a plane!
Also note that dragons already have all the senses they need to fly: unlike humans, who can't feel airspeed, for example. A dragon probably wouldn't need instruments to supplement their senses, only for navigation. After all, people use GPS even when they're walking! A dragon-flight GPS would be more like an aircraft's flight computer, calculating more efficient paths to take advantage of high-altitude winds.
You don't file a walking plan to head down to the grocery store: dragons probably wouldn't need one either. The reason aircraft are required to file a flight plan is due to the collision risk (already discussed above) and the fact there are relatively few flights, and most flights cover a long distance. Assuming that there would be hundreds of millions of flight plans filed per day, most of which cover distances of a few miles or less, the FAA would probably not regulate them merely due to the inconvenience.
Ok, I am trying this without much actual aviation-knowledge, but have been encouraged that this is not out of scope (I actually have no clear idea where IRF-regulations start and end... I know what a dragon looks like, though ^^).
As this answer points out, dragons will most likely not require instruments to see. But they will be safer if SEEN. collision-avoidance would most likely be mainly for the dragons to manage, because they just have an easier time manoeuvring, but for IRF for pilots, it would be helpful if Dragon got Transponders to signal their position.
About weather-minimums: It really would fully depend on the dragons again. They may avoid flying in thunderstorms naturally, and at the same time be perfectly content to start, fly, and land in the middle of the most terrible blizzard... perhaps they just don't get cold, but lightening is an actual danger.
Winds are just as dragon-dependent: if they have capabilities for in-storm-flight like, say, an Albatross, even a strong hurricane may not keep them down.
In bad enough conditions, they may actually have the airspace to themselves!
About what they may sound like to ATC: This is Dragon 101, I am patiently flapping my wings above your tower, may I please land on it until you have the metal-tubes sorted out?
A bit more serious: I would not imagine them much different from normal ATC-talk, just perhaps a bit impatient with those clumsy humans :).
Something that just came to mind about flight plans: we have airways defined for airplanes anyway. Would dragons OUTSIDE these airways be regulated at all? If not, the problem may actually be reduced to having controls in place for "crossing the street" and "I want to land near the airport, please". Dragons purely amongst themselves pose a lot fewer problems that dragons and airplanes.
I think a dragon would still be at the mercy of the FAA.
As with all government entities, they don't like it much when something new enters their jurisdiction, and is unregulated.
They would issue a large stack of amendments to cover operations in controlled airspace (after the review boards, consultants, lawyers, military and other bodies with an interest got through their 'fact finding missions' in the Bahamas).
I'm not convinced that a dragon would have better situational and navigational awareness than say, an AWACS aircraft, which is still required to follow IFR and have ATC clearance in controlled airspace.
Minimum instruments would have to include: Backup attitude indicator: A requirement for IFR at any rate. In order to avoid spatial disorientation. As mentioned before, dragons would likely have a much better sense of motion and balance in three dimensions than a 'seat of the pants' human pilot. But I'm guessing disorientation might still be a factor.
Transponder: With presumably better vision than a human, in a wider range of wavelengths than humans, you're still dealing with airspace full of aircraft operating at a range of airspeeds. ATC would need to be able to maintain separation speeds.
Coms: The dragon would be required to carry two radios.
Radio NAV: ATC:"Dragon Zero One, Dulles Departures. Can you confirm that you are outbound on the two two zero radial?" Dragon 01: "Departures, Dragon Zero One. Negative, I don't have VOR". ATC: "Fail."
TCAS: Not sure what the FAA regs are (I've been out of the game for a while), but TCAS wouldn't hurt.
ASI: Regardless of how well a dragon thinks it can fly, ATC would need it to maintain airspeeds accurately for separation.
Altimeter: Same again. ATC would need to know that the dragon was following a set flight level. Would need it to provide data to the transponder anyway.
Nav lights: Red port and green starboard lights at least. And strobe light. And landing lights.
Another thing to consider: What is the dragon's max cruising altitude? Does the fire breath deplete the oxygen available to the dragon in flight, and at lower air densities? Would supplemental oxygen be required in Class A airspace and/or above 10,000 feet? How would an oxygen rich environment affect fire breath safety?
Other considerations: Weight and balance calculations, and loading charts. As the dragon grows, the weight and balance might need to be recalculated. Possible recertification every year until maturity, then every two years thereafter?
Communication with ATC would have to be in English, the international language of aviation. A dragon would need to be conversant using standardised aviation phraseology. If it's good enough for (or at least tolerated by) the Chinese, French and Russians, it's good enough for dragons.
A dragon is a different beast than an airplane (yes, pun intended).
An aircraft goes from airport A to airport B. It can only stop at airport B (or, when needed, in airport C). Reports are required to metereological service, describing the weather all along the route. The path is seriosly studied by the crew, attending to atmospheric conditions and other considerations.
If, despite that, at mid travel the airplane crew finds that visibility is very reduced, it has two options:
Return to airport A or go back to airport C. Big fuss about time lost, and the passage is not anywhere near B (unless they had luck chosing C). Then deal with the pasengers, refuel, wait for the storm to pass and for the company to assign planes/crew for the new flight. And, when they try again going to B, there is no guarantee that the weather will not become worse again.
Try to flight in less than perfect conditions, expecting that technology will be enough to compensate for lack of visibility.
Now, a dragon wants to go from point A to point B. He gets in the general direction towards B and starts flying. If he finds bad weather, he looks down, finds some clear ground $^1$ and lands. Depending of how tired he is, he will take a nap or he will visit the nearest village, hoping for a maiden to kidnap.
He can repeat it as many times as needed, and with each "jump" he will become closer to his objective. And, in the worse of weather, he can still walk...
$^1$ In case there is no clear ground, burns some houses/fields until there is.