It is not recommended to use a network share for key storage. The best thing to do would be to move the home folder (My Documents>PGP) to a local spot on the machine, then set the program to look there for the keyrings. After moving the folder locally, open PGP Desktop. Select PGP Keys on the left, then in the top menu, select Keys>Keyring Properties, and browse to the new location for the public and private keyrings. That would be your best fix.
If the home folder is on a network share, it should still be accessible most of the time, but latency can affect usage. It will also fail to locate the keys when the system starts, so might take a few minutes before they are able to open anything encrypted. Mapping the home folder as a drive locally can help, but the best solution is definitely to keep the keyrings local.
As far as the key showing as unverified, you should open PGP Desktop, right-click the key, and go to Key Properties. On the properties page, look for Trust. It should be in the left column, 4th option down. Set the Trust to Implicit, and the key should be verified.