Why are all the drains of those AO4406A MOSFETs connected together and to nowhere else? What happens there?
They are connected back-to-back, for different purposes. Charge control, and discharge control. Look up the datasheet for the HY2120 protection IC used, and you will understand how they are used:
The cell voltages are sensed through the VDD and VC pins, and compared to a fixed voltage. Take a look at this block diagram which is from page 5 in the datasheet.
The 2 FETs M1 and M2 are turned on/off depending on what conditions are met.
M1 connected to the OD pin is turned off when cells is depleted. No current can flow out of the battery but because of the body diode of M1, it can be charged up until it is safe to turn M1 on again and removes the voltage drop from the diode.
The same goes for M2. When the cells is fully charged, M2 is turned off so no more energy can be put into the cells, but the body diode of this device makes sure you still can use the energy from the fully charged cells.
Mostly it's for thermal reasons, not electrical. The portion of the transistor that generates the most heat is the drain, followed by the source. On an SO-8 package, the best route for heat out of the chip is the pins, so the drain gets allocated a whole bunch of pins, the source gets almost as many, and the gate gets just one.
If you look at other packages for FETs, like this one you'll see that there's just three connections, but the whole package design is centered around providing a large metal surface area on the drain connection -- this, again, is designed to get the heat out* of the transistor die and onto the board.
* It's certainly not designed for ease of soldering -- trust me on this.