*Will Stripping and Sanding an Existing Finish Affect How the Stain Performs?

Possibly - it will depend on the quality of the stripping and sanding.

If done correctly, and all of the existing finish is removed, the stain will penetrate the same as on raw wood.

If the sanding and stripping are incomplete, the grain and pores of the surface are sealed more, and will not accept stain as easily. More stain may be needed to accomplish the same color as on original raw wood, and longer dry times will be required, especially for thick stains, such as General Finishes Gel Stain.