For better or for worse, a small Beta site sometimes hits the Hot Network Questions list. Is there a way to set up a trigger such that when this occurs, something happens? For instance, I get an email or a ping or something?
There is no such thing built into the network itself. You can probably use the RSS feed for Hot Questions: https://stackexchange.com/feeds/questions and have your RSS client filter for the site(s) you're interested in.