Yes, any lithium ion charger has to stop charging when the battery is full otherwise you'll get a nice explosion, they cannot be overcharged.
WRT discharging all the way, if does nothing to help the battery, they don't need it (lithium ion, or lipo) and it just wastes a cycle. What a full discharge does it calibrate the battery meter. Your % full is based upon voltage and time because the voltage drop over most of the battery discharge curve is so small you can't accurately make a % full from it so they add time used into that equation and base that on previous cycles.
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