Could someone explain to me the speaker configuration on this radio, in particular the output transformer and connection of the ground wires? Is the core itself grounded as well?

The audio output transformer core is physically grounded to chassis. The other coils are not grounded at all. Field coil is part of the power supply filtering circuit. Hum bucking coil connects to audio output secondary coil. (Bucks Hum) This is a typical set up for this vintage radio.