Welcome, Derek.
1. You'll likely tear up the face of the sheetrock when removing the marble and need to replace it with something, anyway. And unless you plan to use Kerdi, you need to replace the sheetrock with CBU, so might as well just tear it all out at once.
2. Aesthetic questions I avoid because you can't see it from my house.
3. From a technical standpoint, I'd leave it alone. It looks like part of the tub. Or consider replacing the tub, too.
4. Need to see why it's there, first. May have been somebody's mistake, or it might be covering some plumbing or some such.
5. You can get generic soap dishes (won't match your tile) in dozens of styles and colors. Or you could build a niche in the wall and tile it to match.
My opinon; worth price charged.