The answer to the second question is "no". Consider the unique solution $x > 0$ to the equation $x^x = 3$. By the Gelfond-Schneider theorem, this number is transcendental. But we have
$$((x^x)^x)^x = x^{x^3} = x^{(x^{(x^x)})}$$
so that two of the parenthesizations coincide. So evidently this set contains transcendental numbers. Lots of other solutions can be similarly generated (e.g., solve $x^{(x^x)} = 4$).