The sound, I think, is coming from a large headset like the ones prominently worn by the ground crew. I don't think it's the pilot's headset, as it does diminish a bit at the greatest range, and I would expect some radio talk between the pilot and the crew. I suspect they are noise-canceling headsets, as there is something about how it sounds that reminds me of vacuuming with a noise-canceling headset on.

Small has been done and cheaper is always possible, but significantly reducing the noise of jetpacks would require an amendment to the laws of fluid dynamics.

As for visible effects of the jet: at liftoff, the tether, that was detached by one of the crew just before, blows around a bit. I am trying to decide if you can see the jet as a distortion of the background, or whether that is an artifact of video decompression. The best opportunity for seeing it appears to be just after liftoff, when the right engine nozzle is seen with the cruise ship behind it, but there doesn't seem to be much of an effect there. As he approaches for landing, the netting around the pad appears to be blown around.