This discussion is getting very off-topic now and will not help the thread starter in any way. (*) Just to answer very quick:
Right. But as most people don’t know the difference, i used the word “battery”. I can change to the correct terms as you wish.
This voltage is then connected on the battery pack including the battery protection circuit, not on the cell.
No, they are not. Look at the bottom part of your circuit. You can see the FETs, where the gate is controlled by the IC. And then consider the voltage drop between source and drain (which should be very, very low, but I don’t know the datasheet of the used component in battery pack of the FP3) and other effects.
And as @AlbertJP already mentioned: You don’t know about the exact materials used in this cell, so you don’t know about the exact charge voltage. If the manufacturer specified it to 4.4V, there is no reason to doubt it.
If you want, you can learn about the basics of charging by just getting some cheap modules from Ebay:
- Get a module with TP4056, which is doing correct CC/CV charging.
- Get a battery protection unit (often called BMS [Battery Management System], even when just used with one cell and no balancing is needed).
- Get a standard 18650 without an included protection unit.
This way you can measure a lot and see how things work. But please be careful and always respect the limits of the cell.
For the thread starter: (*)
Just ask the support to get a replacement 
(*) Not valid anymore as this topic has been split into another thread.