Is the area of a circle ever an integer?

By definition, $\frac{1}{b}$ is the unique real number which, when multiplied by $b$, yields $1$.

By definition, $\frac{a}{b}$ is the product of $a$ and $\frac{1}{b}$.

Since $\frac{1}{\pi}$ is the unique real number that, when multiplied by $\pi$, yields $1$, then $\pi\left(\frac{1}{\pi}\right) = 1$. Hence, $\frac{\pi}{\pi}=1$.

If you allow any radius for a circle, then a circle has integer area if and only if its radius $r$ is the square root of an integer divided by $\pi$, that is, $r = \sqrt{a/\pi}$ for some nonnegative integer $a$.

Your first paragraph, though, is completely irrelevant.

$\frac\pi\pi$ is $1$, because when you multiply $\pi$ by $1$ you get $\pi$.