Do airline pilots ever risk not hearing communication directed to them specifically, from traffic controllers?
There is a ton of radio communication during the taxi, takeoff and departure phases, and then again during the arrival, landing and taxi phases. Pilots are generally very busy at these times and 100% focused. Airlines have two pilots, and one will be flying while the other works the radios and monitors various instruments.

Cruise is another matter. There generally isn't a lot to do other than keep an eye on the autopilot. The pilots will take turns eating, resting, going to the lav, etc. Radio work usually involves little more than getting handed from one frequency to the next. They may get rerouted due to traffic or weather, but it only takes a few seconds to update the autopilot.

Either way, pilots do occasionally miss radio calls. So does ATC for that matter. It's common enough that nobody gets worried about once or twice, though it's annoying. If it's a real radio problem, there are procedures for both sides to follow to keep everyone safe.

There (usually) isn't any sort of light or tone when someone is calling you. However, the standard format is "called station, calling station, message", so with a little training, your brain can quickly tell whether the message is for you. If so, it "tunes in" and you consciously hear the message; if not, it "tunes out" and the message is just background noise.