As Ian said, modern ATC systems will not lose your data from the label if you squawk an emergency code. It'll set off an audible and visual alarm in any ATC center which has a radar feed that can see you. We do indeed train for aircraft doing emergency descents, loss of comms and the other one, as well as setting up fuel dumping and so on. The squawk is far more important than the RT call as it allows sectors below you to deviate any traffic that may effect you before you enter that sector. Even descending at 8-10k FPM from cruise would give a low level or approach sector a couple of minutes to achieve this. If you lose separation with traffic within a few thousand feet then realistically there's nothing we could have done as by the time we get the avoiding action out of our mouth you have probably already descended through their level.