It's true that at the speed of sound, you will have a huge amount of drag. The reason is that the air in front of you has to move out of the way, and if you are moving at the speed of sound, the pressure wave that pushes the air out of the way is moving at exactly the same speed as you. So in the continuum mechanics limit, you can't push the air out of the way, and you might as well be plowing into a brick wall.
But we don't live in a continuum mechanics universe, we live in a world made of atoms, and the atoms in a gas bounce off your airplane. At the speed of sound, you get a large finite push-back which is a barrier, and above this, you still have to do the work to push a mass of air out of the way equal to your plane's cross section with ballistic particles.
As you go faster, the amount of drag decreases, since the atomic collisions don't lead to a pile-up on the nose-cone. But if you look at wikipedia's plot here, the maximum drag at the supersonic transition is only a factor of 2 or 3 higher than the drag at higher supersonic speed, so it is possible to travel at Mach 1, it is just not very fuel efficient.
The speed of sound is constant in the same sense that the mass of an object is constant.
In the typical audible range, at frequencies below, like, $100\:\mathrm{kHz}$ and sound pressures much less than atmospheric pressure, the behaviour of air is very well described by a simple linear wave equation that's purely second order in both space and time. As a result, the dispersion is linear ($|k| \propto \omega$), and phase or, equivalently, group velocity is constant.
Much the same in mechanics: whilst nothing moves anywhere near the speed of sound, momentum and speed are related linearly as described by Newton's laws of motion; we call the ratio mass and it's constant.
Not really though: special relativity tell us that when $v$ approaches the speed of light, you can push kinetic energy and thus momentum ever higher by accelerating an object, but it won't become faster to the same degree. One way to look at this is saying the mass increases.
Similarly, when you go to high frequencies, the wavelength shrinks to a range comparable to the material's internal structure. Therefore, you can't really use a differential equation anymore to describe the spatial component; the dispersion becomes more complicated and the speed of sound isn't constant anymore. Or, if the sound pressure is in a similar range as the air pressure, the behaviour becomes nonlinear alltogether, and you get shock-wave phenomena.
Best Answer
The phenomenon where waves with different frequencies have slightly different speeds is known as "dispersion," because an impulse which begins with lots of different frequencies traveling together will "disperse" and spread out as the faster frequencies move ahead of the slower ones.
Acoustic dispersion is pretty easy to demonstrate. Here's a great video where the effect is audible in cracking ice on top of a lake, and also audible in struck metal cables. Acoustic dispersion in metal cables is how the foley artists for Star Wars created the iconic "pew pew" sound associated with the laser blasters in those movies. In both systems --- ice layers and metal cables --- the higher frequencies have a faster wave speed than the lower frequencies, which why it's "pew pew" rather than "wep wep."
If acoustic dispersion were important in air, then distant impulse sounds --- as from lightning bolts or fireworks explosions --- would also exhibit the "pew pew" effect. Thunder produced from medium-distance storms does tend to start off with high frequencies and end up with a low rumble, but in a chaotic way rather than in the musical way heard in the video for ice layers and metal cables. I think that frequency dispersion in thunder is more about echos off of the terrain and extinction of the higher frequencies in the air rather than the about the higher frequencies outrunning the lower ones.