When you say you need a range of 12-14.4V, 4s li-ion cells will not fit in that voltage range. They are about 4.2V when fully charged -> almsot 17v. So if whatever this is powering can't take that then that is the first problem. It will also droop below 12v possibly depending on how you use it.
So, can you charge 7s in parallel. Of course. 7p is common in electric bicycle battery packs, they are charged in parallel. They are also bulk charged generally (though it is fine to individually charge the groupings as well. This is your other option.)
Bulk charged means you charge the battery as though it was just one big battery cell now and you dont give any heed to their being a bunch of batteries in series and parallel anymore. And individual charging would mean you charge each parallel grouping by its self.
In bulk charging however you need to monitor the individual groups voltages (overcharging these cells is not allowed (leads to fire ) This is why, when bulk charging, you must have a BMS unit to open circuit the charge path and to also balance the cells (not all BMS units balance. Even of the ones that do, most of them do a pretty bad job of it.)
If you want to individually charge them you can buy a charger that will take the bulk leads as well as individual leads of each of the electrodes of the 4 groupings and take care of charing them evenly. It is annoying a bit, but that is your decision to make.
By the way, many cheap BMS units go bad and parasitically drain your cells all the way to zero. The cells are supposed to stop at 3v so this is bad news for the pack. I used bluetooth connecting BMS units so I can monitor the states of the cells and be sure that all is well. They are more pricey though. With higher price you (hopefully) get better balancing functions though.
Oh, of course you will have to charge it with the correct charger (a CC-CV charger is what you need) and you will want to be sure not to charge it too quickly. The cells will have a charge rate on the datasheet) maybe around 1.5amps. 7x that number is fine for your application.
Most of the answer comes from the Operation Flow Chart (Figure 4):
1.1.A) trickle charge will start if the Vbatt Vtc=3 Volts/Cell. So yes trickle charge will start with the cells < 1 Volt.
Note that there is a safety timer that is set using a cap (C10 1uf) on the TMR pin (14). This sets the period of the internal oscillator period to .46 seconds, The trickle charge will go for 8192 cycles or about 3768 seconds , about 62 minutes. If the Vbatt doesn't get above Vtc (3 V/Cell) then it will be terminated and the CHGOK becomes open drain to indicate the timer-out fault.
1.2.A) The recharge threshold only applies when the the charger already charged the battery (Vbatt=VBatt_full and the current came down to Ichg
If you plug the BBB into the charger and the battery is at 4.1 volts, the charger will go right into constant current charge (CCC).
2.1.Q) I agree with you. This seems way to low of a voltage to drop out.
Best Answer
Unfortunately I do not have enough reputation to add a comment, so see this as a Substitute.
A BMS should also monitor the battery temperature (best by a temperature sensor inside the battery pack next to the cells) besides the electrical characteristics. Then you are able to prevent charging in extreme temperature ranges.