Question on the tapping noise, do you only hear it from inside the car or can you hear it with the hood open? If it were a slight rattling noise at low speed, I would have thought it could be the throw out bearing on the transmission. Mine makes noise which I read is normal, so I am totally assuming it would have been the same with a Z-car.
The fact that you've googled the noise on your car and come up with nothing makes it seem like it is not normal.
If others on here are suggesting to walk or have a 3rd party inspect the car, that is what I would go along with.
Anyway, Congrats on the car and getting into a Corvette!!!!! The Z06 is Awe-some! Sorry that it's causing frustration right off the hop.