Calculating displacement amplitude of ultrasonic power transducer

1. Your calculations appear to be correct. I believe the density in your equation corresponds to the density of the material through which your sound wave is propagating. Your usage of steel indicates that you are sending sound through a steel plate (not air). If this is intentional, your calculations are correct.
   You should be using the axial sound speed if your transducer is oscillating perpendicular to the transmission direction. This would apply, for example, if you had a transducer and a receiver, with their transmission axes oriented parallel to one another, both placed on the same side of a long plate. If, on the other hand, the transducer is oscillating parallel to the transmission direction (e.g., at the end of a steel rod, transmitting along the length of the rod), then the longitudinal sound speed is the appropriate speed. (If you intend to transmit through air, then shear waves are not supported. Moreover, in the case of air, the equation you used is not wholly applicable, because the emitting horn is a finite body, rather than a quasi-infinite medium as assumed by the equation.)

2. The value you have computed is reasonable. There are a number of (pay-wall blocked) academic articles discussing transducer amplitudes. These amplitudes range between minimum detectable values of a bit over 1 angstrom (0.1 nm) to rupture inducing amplitudes of ~5 micron (5000 nm). Of course, these values depend heavily on the driving frequency, power, and geometry of the transducer. However, you should have confidence in the values you have computed. I would add that, since the uncertainty in your material density and sound speed is somewhat large, so too will be the uncertainty in your amplitude. To more accurately reflect this uncertainty (using only significant figures, rather than a rigorously computed uncertainty), your amplitude is about $1.7times 10^{-5}$cm.

Edit in response to comment: I understand better now what it is you are doing. The equation you have used above is for computing the amplitude of sound waves propagating through a uniform medium of known density and rigidity (equivalently, known density and sound speed). In the geometry you have laid out, you would want the longitudinal sound speed, because the driver axis is parallel to the horn, which is the transmitting body. However, I must qualify that the use of a small booster and horn invalidates the above result. The result you have computed is the amplitude of oscillations that would be driven, given a known power flux and frequency in an infinite material. This kind of analysis would be reasonably well suited to sending signals through girders in bridges, and finding defects if anomalies in the expected result are found. In the arrangement you are describing, the very point of the booster is to defeat this response by creating a non-uniform distribution of mass thereby modifying the amplitude of the transducer. If you are trying to compute the necessary booster and sonotrode (horn) that you will need, you need to consult the manufacturers specifications as to how they define the driver amplitude. It appears to me that boosters are usually designed around two parameters -- the gain, and the frequency. The transducer, on the other hand, primarily specifies frequency and power. The amplitude will depend a great deal on the rest of the system, i.e., the mass of the horn, and whatever the wedge may be in contact with. While trying to find out more on the subject, I believe I found an answer to your question number 4, although I'm not sure how much use it will be to you. Suffice to say, these types of problems are handled through finite element analysis, not the simple and elegant, but very limited equation you have used above.