Questions about the battery management circuit of Beaglebone Blue
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 &lt; 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.