I haven't been operational for a good few years but I don't think these things have changed much.
For the non-precision approach one, I don't think there's anything laid down but that's what I always did. I suspect there's nothing written down because, technically, I don't think it's supposed to happen - in the world of rules, aircraft are supposed to follow the procedure from whatever fix is specified.
As for the mayday, again, the real world and the rules rarely coincide. Often the first call gave the bones of the problem and I reacted accordingly - I'd acknowledge it but if I didn't need to ask any questions at that moment I waited until the initial rush of stuff to be done (both in the air and on the ground sometimes) is finished. RTF calls - if there was no need to prefix the callsign I usually wouldn't but if another aircraft joined the frequency I'd either prefix the next call to the emergency or give the other a "Standby, emergency in progress".