You can certainly fly into Uluru and fly right out again. But as rwhansen says, that won't give you a sense of that part of the land and culture and historical context. Too rushed, going too far for too short a time. It would be like flying into Yellowstone, looking at Old Faithful, maybe at a bison or two, having a BBQ dinner of bison meat and flying out the next day.
Yes, it's possible, but then so are many things, but possible does not really mean worthwhile. Just imho.
Skip the Gold Coast - I'm sure you can get to beaches with concrete-and-glass highrises behind the sand, and too many chain fast-food restaurants, much closer to home. Nothing special there, actually quite tacky.
Add a couple of days to "The Alice", and a couple of days to Sydney. Take the Manly ferry, walk along The Corso to the open ocean beach, swim between the flags for safety - just as nice as any Gold Coast beach.
Or take the train to Bondi Junction and the bus down to the iconic Bondi Beach, then do the coastal walk down to Tamarama and Bronte and Clovelly and Coogee beaches, take a dip at each beach, now that's good healthy fun!