How many $2\times2$ orthogonal matrices $A$ with $A^3=I$ and $A^2=A^T$ are there? 
How many $2 × 2$ matrices $A$ satisfy both $A^3 = I_2$ and $A^
2 = A^t$
  , where $I_2$ denotes the $2 × 2$ identity
  matrix and $A^
t$ denotes the transpose of $A$?

A bit of manipulation gives me $AA^t=A^tA=I_2$.So this is orthogonal.
Now is there a fixed number of $2\times2$ orthogonal matrices? I have no idea. Please help.

While there are infinitely many $2×2$ orthogonal matrices, there are only two types of them: rotations and reflections. After deriving that $A$ is orthogonal, $A^2=A^T$ is equivalent to $A^2=A^{-1}$ and $A^3=I$, so the two conditions become equivalent.
A non-identity reflectional orthogonal matrix always satisfies $A^2=I$, and so can never satisfy $A^3=I$. Thus $A$ is a rotational matrix, and there are exactly three rotation angles that allow $A^3=I$ to be satisfied: $0$ (the identity), $2\pi/3$ and $-2\pi/3$, yielding the solutions to $A$ as $I_2$,
$$B=\begin{bmatrix}\frac12&-\frac{\sqrt3}2\\\frac{\sqrt3}2&\frac12\end{bmatrix}$$
and $B^{-1}=B^T$.
