Not necessarily. Even the testing for HCV and HIV can produce false negatives. You should always get a second independent test, and a specific test for HBsAg from a highly respected lab. When HCV in particular is dormant or in the incubation stage often testing will show negative results for upwards of 15-20 YEARS, until and if it becomes active. Annual testing is always the best.
No the test for HBsAg is different from Hep C.