I am hoping that someone can help me to calculate voltage and current to the LED socket. I see that there are 3 batteries in series, plus a resistor (?). Thank you very much.
You will have a max open-circuit voltage of about 5V at the LED socket.
As for current, that becomes far more complex & depends as much on your LED as anything else:
The 'forward voltage' of the LED is a fairly fixed value. So, once current is flowing through a given LED, the total current flow is determined by the resistance of any series resistor and the 'overvoltage' of the supply.
"Double A" batteries each have a voltage of 1.5v. There are 3, making the total voltage going through the circuit 4.5v. The current is defined by the load, or what current the LED is able to draw. There's a limit to what the batteries can supply, but any LED you'd put in that socket would be well within said limit. Good luck!