Can a Nikon "D series" lens work properly and let me use the full potential of a Nikon D810?
You can use Nikon D lenses on your D810 with full functionality. In order to control the aperture from the camera you need to set the lens' aperture ring to the narrowest (highest f-number) and lock it there.
But most of the comparable G series lenses are newer designs than their D series counterparts when looking at the same focal lengths and maximum apertures. They usually have improved optical designs with more low dispersion and aspherical elements that help fight issues such as chromatic aberration and distortion and almost always have better lens coatings that reduce things such as flare and more rounded apertures that give better bokeh when used at wide apertures.
Another consideration is Autofocus. Almost all G series lenses are AF-S lenses with built in focus motors in the lens. Many D series lenses are also "AF-S", but many more D series lenses are "AF" lenses with no AF motor in the lens and must be used with a body that has a built in focus motor to drive the AF elements in the lens via a mechanical connection. Although your D810 has a built in focus motor that can focus the D series AF lenses, the in-lens AF motors of AF-S lenses are usually faster and almost always quieter than a lens that has to be driven by the body's AF motor.
So although you can use the D series lenses on your D810, you probably won't get the same AF speed and optical performance as with the corresponding G series lenses. And both of those might be important considerations when shooting with a high resolution camera such as the 36MP D810.