xmas lights bulbs provide continuity but do not illuminate
Christmas light manufactures have started to make mini-bulbs that include a "shunt" device that allows the light to pass current even if the bulb has been burned out.



Image and more detailed information about holiday lights at Energy.gov.

Your "anomalies" are most likely burned out bulbs that have the shunt wire working.  The other bulbs that don't work and also don't pass current have burned out and do not have a shunt, or it is not working.  Neither of these bulbs should be used.

As for the strings that do not work, you can try a device that is made to try and get all the shunts working.  There are products such as the LightKeeper that have a "quick fix" mode.  These send an electrical shock down the wire to try and activate the shunts.  I have used one with some success, but it doesn't always work.

Another procedure I have tried with some success is using a non-contact voltage detector to see where the voltage "ends" on a plugged in set of lights.  I was able to untwist the wire every once in a while on the broken set and probe for voltage.  Using a divide and conquer type search, I was able to find a bad bulb pretty quickly.  That LightKeeper Pro has a built in voltage detector for this, but I think I used another detector I had in the tool box.