When the brakes are checked, what do the firemen actually do?
There are specific protocols that procedurally are required to be followed. Insofar as brake inspection is concerned, it would have been obvious from the trucks that there was no brake fire. It's also obvious that there's not a major amount of smoke coming from the brakes. As they approached the landing gear, they would have smelled serious brake heating. Glowing brakes would be seen then if not sooner. A close visual inspection for very light smoke and sensing the heat coming off the brakes would give them an idea of the rate of brake vaporization. It's a subjective judgement based on what they see and feel.

As I understand it, tires that dangerously overheat will explode before melting, and that's why thermal plugs are used which melt and deflate the tire to prevent their exploding.

At a landing in Nandi, Fiji (short runway) with what turned out to be a tailwind in a 747-200 at near-max landing weight, we had to use max braking, and I called for huffers to cool things down, and of course for an inspection. I exited the aircraft as quickly as possible to see how bad things were. I was amazed and alarmed to see thick dark grey smoke pouring from the body and wing landing gear wheels. I thought we were close to having a fire. However, the ground personnel assured me that there was no cause for alarm, that the smoke was just brake vaporization that would shortly stop as they blew air on them. It did, and after refueling we departed without incident.