If you mean that you have the DVD Player icon on the Dock all the time, if you don't want it there, just drag it off. If it keeps staying there after you use it, then Right/Ctrl click and select Options>Remove from Dock.
Your default system settings will launch your DVD Player app when a DVD movie disk is inserted.
That will make the DVD Player icon appear in the dock, if it is not already there.
When you quit DVD Player (not simply by ejecting the DVD), then the DVD Player icon will disappear from the dock.
More relevant to your question -
No, there's nothing that will appear in the dock that tells you that a DVD is is the drive, unless you have your system set to do that.
A DVD disk (regardless of whether it's a movie, or just a data DVD) WILL normally appear in the sidebar of a finder window - again assuming that you have the system set to show CDs & DVDs in your finder sidebar (from your Finder Preferences.)
The dvd has nothing to do with it. When you run an application, it appears it the Dock. If you wish it to remain in the Dock when you quit, you Right/Ctrl click and select Keep In Dock from the Options tab. If you want it to disappear from the Dock when you quit, you select Remove From Dock from the Options tab.
You can't tell DVD Player to quit and then wait until you eject the dvd before it disappears from the Dock.
Maybe the best way to think of this is showing DVDs on your desktop rather than nav. If you open Finder and then do CMD-, you can opt to show DVDs on the desktop only when you have a DVD inserted. I think the dock is not possible