First question; yes, they should be if your converter/charger is working.
Second question; probably not, only way to tell for sure on either question is to use a volt meter and get a reading not plugged in, then plug the 120 in and check the reading, the voltage should come up some. Then turn off the battery disconnect and the voltage will either start to slowly taper off or remain the same. If it remains the same, it is charging.
99 F250 Super Cab Short Bed 7.3 Powerstroke
84 24' Yellowstone Camino (oldie)