I think you have a pretty good idea what the answers are. I think if you have never been exposed to HBV or have not been vaccinated, then you should be negative for both HBsAg and HBsAb. Actually you should be negative for HBeAg, HBeAb, and HBcAb. If you were vaccinated for HBV, but were not infected, then you should just test positive for HBsAb (negative for HBcAb) - your HBsAb may be below 10 mIU/ml if you did not respond to the vaccine or it may have waned after a long period of time.
You have HBsAb < 3.1 mIU/ml, so this could mean HBsAb is zero. You should ask your Lab for clarification. Maybe the assay your Lab used cannot detect any HBsAb below 3.1 mIU/ml.
If your HBsAb is not zero and you are sure that you have never been vaccinated, then a test for HBcAb may be useful.
Finally, if you have recently received any Immunoglobulin therapy, then the human immunoglobulin used may contain HBsAb.
Just my opinion.
the good news is you are currently not a hepatitis b carrier.
screening for hepatitis core anti-body, if you've ever infected in the past, It will be positive, no need to do anything more, otherwise, highly recommend to take hepatitis B vaccine as soon as possible, the minimum value for anti-hbs is 10 iu/ml to be considered protective.