Your d503 should be fine too, that’s a 13s based board that they gimp to work with 12s. They bridge one of the connections and leave it the wire for it.
You have 13 wires (unless I miscounted) which is perfect for 12s
They do that so they can make 1 BMS and then sell it for different series combinations. I got a 12S BMS that is missing wires as well, because that model can go up to 15s.
Yes but the charging current is going through those mosfets as well, if you would use diacharge on it it would also go through them but since your bypassing it the discharge current is going directly from the battery.
By the size of it the tiny bms looks to me that it does not have the balancing feature, but I would doublecheck that on the website
As the smallest bmss usually do not have it