I would check this link. https://circuitdigest.com/calculators/lm317-resistor-voltage-calculator It's just a LM317 calculator.
I think 10k potentiometer is a little much. Max for the potentiometer should be 2400 ohms to get 13.75 volts.
Or just replace R2 with 1K ohm and it should be fine.
I don't think it matters either way to have the battery disconnected or not, might help to be on the safe side though and have a Volt or Amp meter alongside this circuit.
The zener's breakdown is rated for 11 volts that's when the Led will come on.
I believe 15 should be fine, what is the amperage?
There are better circuits out there, but this one will get the job done if the values are correct.
No. The value of R2 must be low enough for the LM317 minimum load current which is a max of 10mA. Therefore 1.25V/10mA= 125 ohms, use 120 ohms. The datasheet shows a 240 ohms resistor for the more expensive LM117 that has a minimum load current of 5mA.
I don't think it matters either way to have the battery disconnected or not, might help to be on the safe side though and have a Volt or Amp meter alongside this circuit.
No. The datasheet for the LM317 says its input must be at least 2V higher than its output. The battery label said charge at 13.7V and there is a diode and resistor at the charger output so the output of the LM317 is 15V then its minimum input must be 17V.