When I have automatic docking configured in the client settings and approach a dock, it occasionally displays "Docking," but nothing happens. I press the Activate key, and the ship docks as expected. This occurs at various stations, but it seems to happen more frequently at locations like Pelatus C-12. Could it be due to fog or lag?

For me, this occurs when I undock from something and try to re-dock too quickly. Either auto-dock has a cooldown (prevents in+out dock ports from getting you stuck) or the events that enable it are being delayed a bit, or something.

Since you mention Pel's conq station, I assume you're loading a capital vessel via shuttling? Try increasing the spacing between you and the station a bit.