Is there a DAC (Digital to Analog Converter) in the iPhone Lightning Dock? I'm wondering if there's a DAC in the dock because the first sound loudness level is noticeable higher than the first level from iPhone 6+ when the iPhone is mount on the dock.

I suppose if there's a DAC that it has a lower resolution, i.e. less steps between silent and max loudness making each step wider on the dock.
Yes, because the Lightning port is digital-only. The Apple Lightning to 30-Pin Adapter also contains a DAC, according to MacRumors.

I don't know about the quality, but there is much more space for a good DAC in the dock then there is in the phone, so I wouldn't say it has to be worse.