Yes, original battery.. I will check Voltage when possible.. currently works in Camera but unable to charge seems strange...Is it the battery in question the original which came with the camera?
Running a lithium ion cell down all the way ought not fix anything.
Do you have access to a digital multi-meter? if so, measure the open circuit voltage across the terminals. A fully charged cell (the NP-40 "battery" is actually a single lithium-ion cell so technically is a cell, not a battery) will be somewhere around 4.15 - 4.18V when fully charged. Probably the camera will not leave the cell discharged more than 3.5 - 3.6 (measured at rest, not straight out of using the camera). I say probably because I'd have to examine the camera to know for certain. The nominal voltage of 3.7V on the cell is not the fully charged voltage, by the way.
If you have a meter, check it (be careful not to short the cell - that is very dangerous) and let me know what the reading says. If unsure, don't do it.
No idea, but get a few extra - they're quite inexpensive and I need at least two for a full day of shooting with this cam so far. I have three and have never QUITE needed the third but was right at the end of the second at the end of the day once...