Metric Space Question: is $H(x)$ in this neighborhood?

This metric is actually quite easy to visualize. Draw the graph of both functions and then the largest vertical distance between $f$ and $g$ is exactly $d(f,g)$. In this case the largest distance occurs at $x = \frac{1}{2}$, so your answer is correct.

$d(H,f) \ge |H(\frac12)- f(\frac12)| = |0 - \frac12|= \frac12 $ as the sup of a set is an upper bound for it. So $f \notin B_{\frac12}(H)$ as claimed. Well done.