Also about the external hdd, how can a full erase correct the pending sector count, even if temporarily?
Bad sectors can be soft or hard. Soft bad sectors might be temporary. There are different scenarios here and depending upon the situation bad sectors are treated differently.
When disk sector become bad, it may reflect under Current pending sector count, as what was seen in your first snapshot. This means that the sector is bad and is waiting to be remapped. However firmware has not yet reallocated it. Here since it's not a good sign, disk status becomes Caution. If after a few more attempts the sector is read properly (mostly being a soft issue) then it need not be remapped and is reversible. Most of these propitiatory tools (as well as third party tools like HDD Re-generator) attempt to read again and again from the bad sector to see if it can be reversible. If it works well then pending sector count decreases, reallocated count remains unchanged and if there are no more pending sectors, the value is restored and disk can in fact become from Caution to Good as what is seen in your next snapshot.
On the other hand when the sector is bad and can not be read anymore, firmware reallocates it to the spare area and then the Reallocated Sector count increases and disk status becomes Caution.
Generally reallocated bad sectors are not reversible and disk status does not change from Caution to Good. However in case of soft issues the process may be reversible.
I had a similar issue with my Seagate laptop HDD. It developed 96 pending sectors. I took a snapshot and mailed it to their RMA team. Before however sending the HDD to the service center, I did a low level format and everything was restored perfectly. All 96 pending sectors were recovered. I still decided to RMA it and got a replacement though.