Do autopilots in non-glass cockpits feed off of the gyro instruments? Do autopilots in non-glass cockpits feed off of the gyro instruments to determine aircraft attitude, heading and rate of turn? Or do they have their own internal gyros? Since I can set the heading using the heading bug, I suppose it needs to be able to read the heading-bug offset at the very least, so there's some connection.
Yes, they do. Also for glass cockpit, the autopilot uses the sensors that are used for the displays (ADC, AHRS, etc).
Here are some extracts from the FLM of the SA 365N Dolphin, describing the autopilot and coupler (I selected only parts relevant to the question):