That can happen. However,
the ELISA test (IgG) is more sensitive than WB. If
ELISA is negative, positive WB is rare. For that reason, WB is not normally used to test people with negative IgG tests for HSV-2. Its primary purpose is to confirm a positive ELISA, especially if that gave borderline results.
So the direct answers to your questions are:
1) As far as known, there is no such thing as a false positive WB for HSV-2. A positive WB always reliably indicates the person has HSV-2.