My preferred introduction during the course of my extensive experience has been:
G'day, I'm Melclin, this is my partner Barnaby Joyce, what can we do for you.
At uni, people will often introduce themselves: "Hi, I'm Esmarelda from the Ambulance service." and I always think.. they've called 000 for an ambulance, you've turned up in an unambiguous uniform with paramedic stamped all over it, carrying a bunch of medical stuff and there's a bloody great ambulance parked in the driveway... I think they're ganna get that you're from the ambulance service. Haha. (I do realise there may be many good reasons for announcing yourself but I just thought it was a mildly humorous observation).
While I don't know that you could say that any particular question is the MOST important, it seems to me that one of some considerable importance is the kind that queries the normality of the presentation, "Is this how your chest pain normally is", "Is this what Kevin is usually like during his psychotic episodes", "Is Barry usually a quiet baby".
Also, I'm terrible with names, a trait that I'm trying to improve, but for me, male patients are 'mate' and female pts are 'love' until proven otherwise. Mothers are 'Mum', Fathers are 'Dad' and anyone smaller than me is 'little tacker'. B)