@movonw You might wish to connect a power supply instead of the battery instead (buffered by a large capacitor, because the phone surely relies on the battery to deal with power usage surges).
@movonw I would avoid trying to charge it. If you want to, use a specialized charger for Li-Ion batteries (for example https://www.adafruit.com/product/1904). Incorrect charging of such batteries (either too high current, or too much overcharging) can cause overheating and fire (and I don't know where the ultimate protection circuitry for that battery is located -- on the battery or in the phone). It's not that hard to do the correct thing yourself (it's mostly a matter of restricting current as a function of voltage), but figuring out if the thing you're doing is correct is less trivial.
@robryk Thanks for pointing that out! I am a noob wrt to electronics, I will ask someone who knows better for this. Perhaps a repair shop would have such a specialized charger.