Input Voltage range for Wemos D1 board
It's 3.3V linear low drop out regulator has a drop out voltage of up to 0.3V.

The 5V regulator from VIN is unlabeled in the schematic so we can't say. Depending on how much it drops at 4.8V, the 3.3V should still be powered. But at 4.8V, your 4 AA pack is dead anyway. 

The board uses a comparator to disable the USB power input when the DC jack is used, so no,  it would not work that way without you disconnecting the battery pack.

As listed on the website, it says the DC input is 9V to 24V. There is a SS24 diode in the way, so that will drop about 0.3V at the currents used by the Wemos. Again, without knowing the 5V LDO's drop out voltage, then it may need much more than 6V. 

Frankly, you should inject the batteries to the 5V rail, through a 1n400x diode. That would work nicely, as the 5V is only used to power the 3.3V regulator.