I can see that it could be wise to pull the emergency handle at the station to prevent the train moving if, say, someone were having a heart attack and couldn't easily be moved to the platform.
But pulling it between stations would only prolong the time it took for the person to get help.
It won't as there are no emergency brakes on Tubes, they're just called emergency brake. The train continues to the station and will then remain there.
Regardless of whether the emergency handle is connected to a physical brake or if it sounds a alarm, there is an emergency brake. If the driver removes their hand from the dead man's switch, the emergency brake engages. But that's a side-road to this discussion.
Unless you can point to official advice from TfL that states otherwise, I'm just gonna come out and say that I think it's best to stick to the advice TfL give at stations which is not to pull the emergency handle if someone is unwell. Whatever the precise action that results from pulling the handle (whether that's sounding an alarm, engaging a brake, or whatever), I don't see what it would achieve between stations and especially when TfL say not to do it.
Yes, but the audio quality is not very good. There was a recent incident that was exacerbated by a driver mishearing "there's a fire" as "there's a fight".
I think they started these announcements because people considered someone feeling dizzy a medical emergency.