As you heard it on centre it was presumably for an arrival.
The "Heavy" suffix to ATC isn't dependant on the actual weight of an aircraft, only which category the aircraft fits in to at MTOW. So for Emirates they're always either Heavy for a 777 or Super for a 380.
Refusing to accept 27 in Melbourne could be to any combination of factors. The aircraft may have dispatched with a thrust reverser out, auto speed brake out, a brake locked out etc, all of which affect stopping distance. Or perhaps they were close to MLW and didn't want to land on a limiting runway after being up all night, when there's another option that is 1400m longer.
At the end of the day the decision rests with the PIC, and ATC are there to pull the resultant "spanner" out of the works if the duty runway is deemed unsuitable.
