I wired a GFCI outlet outside for a Koi pond pump, buried the 12-2 Romex® wire in PVC pipe and ran it to the house and into my circuit panel on a dedicated 20 amp breaker. The house is for sale now and the home inspector has a problem with only the outside plug being on that breaker. What is wrong with my installation?

NM cable is not allowed to be installed in conduit, or in a wet or damp location. Also, the receptacle should have a weatherproof in use cover affixed to it.
Otherwise, as long as the receptacle is rated for 20 amps, there’s no reason why it can't be on a dedicated circuit.

I have a new above ground pool. I was informed there is a new code for the electric for the pool pump motor specifying a #8 copper bare wire to go around the whole pool as well as other changes. Is this correct?

The electric code for swimming pool installations is contained in part 680 of the NEC and is quite involved. It is not new, but as with all sections of the Code, it’s subject to updating every three years. You have to check with your local building department or other AHJ to find out which version is in force in your community. Ultimately, that person will have final say as to what he or she will allow in your town.

Essentially, this code dictates what type of wire is allowed for powering your pool pump (you can't use just any wire), equipotential bonding grids, convenience receptacles, and the proper installation of GFCI circuit breakers.
We usually do NOT recommend that novices attempt to do pool wiring; since you're combining electricity and water (in which people will be immersed), you want to be sure that the installation will not lead to a fatal shock sometime in the future.

Lastly, keep in mind that wiring for a pool can tend to be costly. This is due to the fact that there are some rather expensive requirements that your licensed electrician must adhere to for your safety. We always recommend that you get at least three estimates for the work to be done. Most people tend to shy away from the higher cost estimates; we would suggest that you should also question any estimates that seem like a bargain when compared to the others. Chances are this person is not quoting everything required by Code to ensure your pool wiring is safe for when you are your family are swimming!