As for why the viewport changes when the eyedropper is visible, this is because of the tonemapping of the viewport. If you have the default tonemap of ACES applied, when enabling the eyedropper, the viewport automatically switches to sRGB and to an exposure of 1. To sample accurate colors from the base color without lighting, you can press C to enable solo channel mode and view the base color channel. You can then enable the eye dropper, where the viewport will again temporarily set the tonemapping of the viewport to sRGB and an exposure of 1 to accurately pick the color off of the mesh.