How to not damage leather while skiving

Ah, I did wonder if it was the edge of the paring stone. I used an orbital sander on the edges of mine to avoid this. It may be worth giving yours a power sand.

As for recovering the leather, probably not as you're using aniline dyed vegetable tanned leather in a lighter colour. Just about the worst type of leather to fix with pressure, heat, moisture or surface pigments.

If the bag is unlined, you could use a pointy object right behind the dents to somewhat push them out. This may help a little, but won't fix scratches.