Identify the y-intercept.

B is the y-intercept of the function $g(x) = a^x$. To find the y-intercept, set $x=0$.

The coordinates of B are $\boxed{\text{} (0, 1) \text{}}$

Point A $(-3, \frac{27}{8})$ lies on $g(x) = a^x$.

Take the cube root of both sides:

The value of $a$ is $\boxed{\text{}\frac{2}{3}\text{}}$

A reflection in the y-axis means replacing $x$ with $-x$ in the function.
Given $g(x) = (\frac{2}{3})^x$.

The equation of $h$ is $\boxed{\text{}h(x) = \left(\frac{3}{2}\right)^x\text{}}$

The equation of $h^{-1}(x)$ is $\boxed{\text{}y = \log_{\frac{3}{2}} x\text{}}$

Since the base $b = \frac{3}{2}$ is greater than 1, the inequality direction remains the same.

The argument of a logarithm must be positive.

Combining the conditions, the values of $x$ are $\boxed{\text{}0 < x \leq \frac{3}{2}\text{}}$

The domain of $h^{-1}(x) = \log_{\frac{3}{2}} x$ is $x > 0$.

For $h^{-1}(x - 3)$, the argument of the logarithm is $(x - 3)$. This argument must be greater than 0.

The domain of $h^{-1}(x - 3)$ is $\boxed{\text{}x > 3\text{}}$