If two sound waves that are different frequencies create beats that occur several hundred times per second, can you hear this effect as its own tone? If you have multiple waves of different frequencies, the interference from the different waves cause "beats".
(Animation from https://en.wikipedia.org/wiki/Group_velocity)

Let's say that a green dot in the above animation reaches your ear a few hundred times per second. 
Is it possible to hear this phenomenon (wave groups occurring at frequencies in the audible range) as its own tone?
 A: Hearing 'beats' at a frequency n, as in the above example, is not the same as hearing a note at that frequency. In the example you give, there is no actual note present at the lower frequency, i.e. the air is not being excited at that frequency. All you are hearing is an interference effect at frequency n. For example, if you were to convert that example waveform to the frequency domain (i.e. spectral analysis), you would see two higher-frequency spikes very close together, but there would be no spike present at the lower frequency n.
Your ear would hear and interpret the interference effect as the volume of the note increasing and decreasing at the frequency n. This effect can be used, for example, when tuning a guitar string - play two notes that are supposed to be the same on two different strings simultaneously and you will hear beats if they are slightly out of tune.
If you were to superimpose a lower-frequency note on top of a higher-frequency note (i.e. two notes played simultaneously), the waveform would look quite different (more like a high-frequency wave 'riding' a lower-frequency wave, as shown in the image below). In that case, your ear would hear the two different notes simultaneously.

A: No, one cannot hear the actual beat frequency. For example, if both waves are ultrasonic and the difference in frequency is 440 Hz, you won't hear the A (unless some severe nonlinearities would come into play; edit: such nonlinear effects are at least 60 dB lower in sound pressure level).
When two ultrasonic waves are close in frequency, the amplitude goes up and down with the beat frequency. A microphone can show this on an oscilloscope. But the human ear does not hear the ultrasonic frequency. It is just silence varying in amplitude :)
(I know a physics textbook where this is wrong.)
Edit: in some cases the mind can perceive the pitch of a "missing fundamental". For example, when sine waves of 880 and 1320 Hz are played, the mind may perceive a tone of pitch A. This is a psychoacoustic phenomenon, exploited for example in the auditory illusion of an Escher's staircase.
A: Yes - American Technology Corporation, Woody Norris invented a phased array consisting of ultrasonic transducers; pairs that transmit two ultrasonic frequencies that are slightly different by a modulated sound frequency. 
Demodulation of the audible signals from the ultrasonic carriers is accomplished either by nonlinear properties of air or by the two signals striking a surface such as a wall or the inside of your head! In any event the sound appears to occur virtually out of thin air.
These devices have been called hypersonic sound speakers or audio spotlights. Once in awhile you can find them for sale on EBay
A: As always for anything involving biology, the answer is actually more complicated. 
It is true that there is no "note" there at the beat frequency, in terms of Fourier series. But despite what is commonly stated in textbooks, the ear does not just do a Fourier transform.
In fact, the human ear does perceive differences in frequencies, and more generally certain linear combinations of frequencies, as actual tones. They are called combination tones, and a demo is here. As you can hear in the second clip, when two frequencies $f_1 < f_2$ are played, one hears tones at frequencies $f_2 - f_1$ (the difference tone) and at $2f_1 - f_2$ (the cubic difference tone), as well as some others. This is no small effect; these tones are several octaves below the original tones. 
This would be impossible if the ear were a simple linear system, because there is no Fourier component at frequency $f_2 - f_1$ or $2f_1 - f_2$. But the ear is nonlinear, and its output is then subsequently processed by the brain, again in a nonlinear way. And it's well-known that the simplest thing nonlinearity can do is output linear combinations of the input tones; that is one of the cornerstones of nonlinear optics.
While the theory is not completely understood, almost everybody can hear the difference tones are there. However, in the case of extreme ultrasound, it's quite unlikely that you'd hear anything because an ultrasound wave can barely budge anything in your ear at all. If your ears are not sensitive enough to detect them in the first place, it's unlikely they would be able to output nonlinear combinations of them no matter how nonlinearly they process the sound.
A: The human ear detects sounds by having hairs with different fundamental frequencies; if an incoming frequency is sufficiently close to harmonic of the hair, the hair detects the sound. In essence, the ear performs an analog Fourier transform. While the graph of the beat looks like a sine wave, its dot product with a true sine wave is zero, thus it's not detectable.
