1. The underside could easily be cleaned and polished. You could try a mild acid (e.g., vinegar, perhaps with baking soda) to remove the brown ring which is likely from furniture polish. If you want a high polish (and it doesn't have one hiding under the patina of ages) a decent gravestone maker could probably do it -- or try Bon Ami.

2. The gouge was probably caused when someone dropped something heavy on it. The beveling on the edge is also damaged and the edges of the damaged area are weakened and could eventually fall out. You'll need a good workman to repair it and he wouldn't guarantee the results.

This from an old (years ago) gravestone maker.