Wow, a lot of questions! I'll answer what I can here, but first off, it is well worth checking out PBS Nova's documentary Faster than Sound which covers the X-1 and the race to break the sound barrier. It features interviews with the participants and a good technical summary of this early generation of supersonic test aircraft.

Numbers 5. & 6. The big competitor to the X-1 was the British Miles M.52 and the de Havilland DH 108. Although there was some testing that showed significant promise, the Miles design fell victim to Britain's tightening postwar budget which used concerns about pilot survivability as a pretext to eliminate the project. The Swallow was in the words of Eric Brown, Britain's premier test pilot, "a serial killer." The de Havilland machine was tricky to fly and quite dangerous. The Bell team succeeded in part because the team had funding and was willing to accept design compromises. The fact that Bell's comparatively lackluster wartime production meant they had the shop space and designers available when the Army Air Corps pushed for a manned supersonic rocket aircraft in 1943. The thin straight wing represented Bell's pragmatic approach to design. In the mid-1940s, it was conventional wisdom that the optimal design for supersonic flight was a swept-back wing, and while this was true, many design firms had little experience with swept-back wings. The notable exception to this was Germany, whose firms had pioneered a number of experimental concepts. So a number of aviation firms sought to instead try and comb through German data and use the expertise of German engineers before proceeding in their designs. Bell instead opted for a straight wing. De Beeler, one of the engineers involved in the X-1 program summed up Bell's approach vis-a-vis the British

Without question, we actually had more information on the straight wing. So, if you sized up the various advantages and disadvantages, we had the smallest degree of risk to enter an area that we didn't have all the answers for, to go with a straight, thin wing. Frankly, I was a little surprised that the British didn't go ahead, because I have a good feeling they could have made it. With all due respect to the people responsible for the cancellation [of the Miles M. 52], I think they made a mistake.

Number 12. This comes from an earlier answer about what would have happened had a Cosmonaut landed on NATO territory.

The legal position of spaceflight and its associated emergencies is a very nebulous one in international law. The advent of manned spaceflight superseded a number of laws regarding the legality of overflight and the nature of sovereign territory. For example, the 1944 Chicago Convention helped establish the groundwork for International Civil Aviation Organization- ICAO which clarified a number of the basic rules for emergency landings and what constituted an infringement of national airspace. The ICAO established that the airspace above national boundaries is national territory, there can be no neutral crossing of this territory. Space vehicles sidestepped this legal precept. De Gaulle demonstrated this facet of outer space to Khrushchev when the latter went on a harangue about the Gary Powers U-2 incident at the Paris Peace summit. He told the Soviet premier "yesterday that satellite you launched just before you left Moscow to impress us overflew the sky of France eighteen times without my permission. How do I know that you dd not have cameras aboard while taking pictures of my country?" upon which Khrushchev flubbed his answer. As laws regarding space have evolved, a general consensus has emerged that space is neutral territory, unlike that of the atmosphere.

As space vehicles transitioned between atmosphere and space, they occupied an indeterminate zone. The UN sought to clarify matters in the late 1960s with two international treaties that outline both the responsibilities of the state that launched vehicle and the wider international community as a whole. The 1967 Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies maintained the idea that astronauts and space exploration as a whole was a human endeavor that transcended national boundaries. Later, and more pertinent to the OP's question, the Article 4 of the UN Resolution for Agreement on the Rescue of Astronauts, the Return of Astronauts and the Return of Objects Launched into Outer Space established that in case of emergency, all signatories of the resolution would:

If, owing to accident, distress, emergency or unintended landing, the personnel of a spacecraft land in territory under the jurisdiction of a Contracting Party or have been found on the high seas or in any other place not under the jurisdiction of any State, they shall be safely and promptly returned to representatives of the launching authority.

Of course, these UN resolutions are dependent upon self-enforcement and whether or not either superpower would have upheld them is venturing into what-if territory. However, it is likely that they would have obeyed the strictures of these late 1960s agreements as they fit within the established paradigms of international rescue pioneered by the ICAO.

Why were the wings so thin?

To maximize the lift to drag ratio at supersonic speeds. While it might seem obvious that thinner means less drag, the purpose of wings is to provide lift, so the lift to drag ratio is what counts. For subsonic flight it's a teardrop cross-section, but for supersonic flight it's a very thin wing (typically with a diamond shaped cross-section).

The X-1's fuselage shape was based on that of a supersonic bullet, so it was known that the sound barrier isn't a "brick wall."

A bullet is very different from a plane, because it lacks wings and control surfaces (rudder, ailerons, etc.). The main concern in going supersonic was being able to keep the plane stable when it went from subsonic to supersonic flight. The part in between is called the trans-sonic region. At subsonic speeds the air that the plane pushes out of its way goes in front of the plane, because the displaced air travels at the speed of sound (sound is essentially a series of air displacements) which is faster than the plane. At supersonic speeds the displaced air travels behind the plane because the plane is going faster than the speed of sound (kind of weird if you think about it). You can design things to work in either region, but in the trans-sonic region it's neither here nor there and leads to instability.

I don't know how historically accurate that part is, and it's a long time since I've seen the movie, but that would be why "Yeager [thought] that the buffeting would lessen, as Mach 1 was approached".

For that matter, NASA describes the X-1 as the first plane to go supersonic, in level flight, suggesting that another plane had previously traveled at Mach 1, just not level to the ground. What plane and circumstance was that?

As /r/gantimatter explains, no one is certain if any previous planes went supersonic. If they did it was in dives, so saying "in level flight" is a record that couldn't be challenged.

NASA describes the X-1 as the first plane to go supersonic, in level flight, suggesting that another plane had previously traveled at Mach 1, just not level to the ground. What plane and circumstance was that?

There were a few claims by WWII pilots that they'd passed the sound barrier by going into dives.

One of the more well-known is Hans Mutke, although he only made his claim years after Yeager's flight.

Why did my diving Me 262 Swallow get really rough, then suddenly go all smooth, then get really rough again as I decelerated? Sounds like what Yeager's plane did...and my speedometer was maxed out, so....

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What criteria, exactly, were Mercury volunteers evaluated for? How were the criteria devised?

NASA has a great archive describing how the first astronauts were selected - some of it overlaps with some of the stuff about psychology and history currently on /r/AskHistorians' front page.

Of note to me at least - some of the first candidates were eliminated because they were too tall.

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Who was the first non-Protestant US astronaut?

Gemini 4 commander James Alton McDivitt was the first Catholic astronaut.

I'd speculate that all the Mercury astronauts were Protestants (on paper, at least - I'm not sure how devout all of them were) because that's a pretty accurate sampling of mainstream America at the time, or at least the America with bachelor's degrees in science and engineering. Catholics were a minority. That's just a guess, though.