People's assessment of noise is subjective, so it is extremely hard to be definitive about your problem.
As an observation, the NSX drivetrain seems to have a largish amount of lash in it. The 6 speed gearbox seems to be a little worse than the 5 speed in this respect. The OEM clutch on the 6 speed box has a relatively narrow engagement point on it which makes smooth jerk free take offs a bit of a challenge. The lurch, near stall, disengage, rev the engine, pop the clutch again kind of start seems to moderately common with new owners. Aftermarket performance clutches just make this problem worse because of typically narrower engagement points. My observation about some owners is that they deal with the lurch / stall problem by revving the engine up to around 2000 - 2500 RPM and then slide the clutch out. Gets you a very quick stall free start, short clutch life and perhaps a rear ender into the back of the car ahead of you if its rush hour traffic. Getting a smooth start without starting with a 2000+ RPM engine speed requires some finesse.
Do as suggested and check the front and rear engine mounts to confirm / eliminate them as a source of the problem. The front one is a hassle to get eyes on without removing stuff - one of those remote inspection cameras might help. You didn't specify the year / mileage on your vehicle. A high mileage older car is definitely a candidate for worn mounts, particularly if it has been enjoyed vigorously by previous owners. A completely failed front mount could certainly materialize itself at a bang at the back of the seat region following a 1-2 shift followed by full throttle.
If the mounts seem OK, then jack one rear wheel up off the ground and with chocks to prevent movement of the vehicle, the transmission in gear, emergency brake off and the wheel removed, use a bar between the wheel bolts to rotate the wheel back and forth. This may allow you to determine the location of the noise. A lot of rotation in the hub would be unusual. I did this on mine; but, its been 6+ years so I can no longer give you a quantitative number for the degrees of free play. Repeating left to right may allow you to assess the relative condition of the L/R driveshafts. Of course, if you spot split driveshaft boots or ooze from the boots you could have an answer to your noise. Not sure about your reference to U joints since the NSX does not have any. Do you mean the inner and outer tripod joints on the driveshafts?
If your car is still moving, it is clear that you have not had a transmission or differential failure. Wear in the differential will certainly increase drivetrain snatch; but, usually comes with continuous noise / howl telling you there is a problem. Same deal with gearbox bearing wear and its usually the synchros that become a problem first in the box short of outright knocking some teeth off of a gear pair! That would definitely make continuous noise. You can try a drain and flush on the transmission (always good if you don't know the last time it was done). If the fluid comes out with a bunch of shiny metal bits in it, then you have a transaxle issue. If the fluid comes out clean then chances are that the wear levels are probably normal.
If you list your location, you may be able to find another local Prime member who can meet up with you and go for a short drive in your car. This could give you a quick read on whether your noises are abnormal.
