Well, I have found that my peace lily likes a fair bit of water. I let it droop only very, very slightly and then water, but then it springs right back upright again. Brown edging on the leaves of a peace lily are caused either by too much water or not enough water. If you say you let it dry out enough to wilt, then it could be you're letting it wilt too much before watering. The one picture you posted there is rather severe wilt. Don't let it wilt even a third of that much. The leaves should only be showing a hint of a sag and then it's time to water. Make sure your pot has drainage and water just enough to have water come dribbling out the bottom holes of the pot, let it drain fully, and dump out the water from the bottom tray. It doesn't like to sit in water. Also, a peace lily does not want direct sunlight or really bright light. I actually have mine sitting in a corner away from the window and it blooms happily there. They say if the flowers turn green it is getting too much light. That said, it's not one for total darkness either. Off to the side of a north window is really good.

I know it doesn't seem possible, but a 10" pot holds about 5X (4.8) as much soil a 6" pot, so be careful you're not over-potting. How I would treat the plant: First, I'd make sure I have a slightly larger pot ready to go with screens over the drain holes and a 1" layer of potting soil mixed 50/50 with perlite covering the bottom of the pot. I'm only recommending that soil mix for this application because it should be compatible with the soil the plant is in. I'd prefer something with more aeration that drains better than that, but you shouldn't use 2 soils that are significantly dissimilar in one container. I'd then use a utility knife with a fresh blade or a new single edged razor blade to cut 4 vertical slits 1" deep in the root mass from top to bottom, then I'd cut or tear off (with my hands) the bottom inch of the root mass, set the plant on the bed of soil, and fill in around the perimeter with fresh soil and water lightly. After a couple of waterings, I'd fill in again around the edges, making sure I don't add enough soil to cover the top of the original soil/root mass. Don't worry - your plant won't balk at this treatment. Alternately: you could just hold off until the plant starts telling you it wants to grow, (prolly around Easter) then divide it with a butcher knife or similar tool, pot the divisions, and share your largess. Al

Photo is not sharp enough for me :(, but am wondering about the stems: they look too 'soft'/fleshy for a fern...(but I know there are many ferns, and really can't be sure looking at the photo)

They're likely tiny particles from the inner part of the plant's flower, the stubby part (spadix) inside the whitish leaf-like bract (spath). It's entirely normal. I hope that's not a self-watering pot, as they can be trouble.