have a look at the schematics of M5Stack Core and PLC.
I do not have the PLC module but I do have the LAN module which is similar in that respect and I do have both 12V and USB connected at the same time while programming the M5Stack Core, but not permanently. Both power sources produce 5 volts but there might be a slight difference in the voltage which would mean one power source is feeding a small current into the other, but I don't think it will damage anything. Nevertheless, use at your own risk.
The full project, M5Stack Core and PLC module will work fine from USB only. Unless you add something in the PLC module which requires 12 volts.
According to the PLC module schematics the BAT pin is not connected / used. But it will show the battery voltage from the stacked M5Stack Core.
No, the battery in M5Stack Core will not charge when only 12V is applied. Only USB can charge the battery.