Do I need OD/OC/OV Protection for Small LiPo Batteries with a Protection PCB already?
It is a good safety practice that you should always have at least an extra layer of protection when dealing with Li-ion batteries.
A proper design shall have:

A well designed, robust charging and discharging circuit
A battery protection circuit
A battery with protection circuit built-in

#3 would be the last resort in case both 1 &amp; 2 failed.
You can easily find many designs in China skipped #2, and some I have came across even without #3 and a poorly designed #1, and with a crappy power supply. Not too difficult to imagine how these would ended up after being used for a while.