When you trim the damaged leaf, use sharp scissors to cut so that the damaged portion is removed, but don't use a straight cut. To make it look better, trim the leaf so that the original oval contour is maintained. It's like trimming your fingernails. This trimming is only about aesthetics and not at all about the health of the plant. Just trim it (or not!) so that it looks good to you.
I don't see any unusual wilting in the photo you posted. Peace Lilies do wilt dramatically whenever the soil gets just bit too dry. Fortunately, they do perk up again several hours after they are watered.
Yes, you can rinse the soil and move it to a container of water. But why would you want to? It will take the plant a while to make the transition and you will lose some leaves in the process. In addition, it is likely that you have more than one plant growing in the soil. I just don't see any advantage to moving it to water.