1. Cut the yellow leaf off at the soil line. It will never fall off by itself, and it's ugly, so why leave it.
2. I guess you're talking about the water in the clear plastic liner. First, dump it out, then let the plant drain for a few hours - you can leave it in the liner - dump it again, and continue until you have no more than 1/4" water in the liner. Now let the plant's soil dry out - we call it aerating - until the leaves just begin to bend down on their stems. This is how peacelilies tell you they want more water. Water it slowly until you have 1/4" in the bottom of the liner, and she should be good for a week or so. Peacelilies will die if their roots are kept too wet, but they're good plants for beginners, because those bending leaves tell you right away what's going on. You can learn about how the soil should feel by digging up a little with a spoon, and squeezing it between your fingers. Try it when the soil is very wet, like it is now, and a week from now, and when the
leaves start to wilt; also try the soil on the surface, and deeper into the pot.
3. Don't repot it. Its grower has given it good soil and fertilizer. You don't have to repot at all, until it gets too big for the pot. Instead, get an attractive container that will hold plant,grow-pot, and liner, and set your peacelily into it. You can put some spanish moss on top to cover up the grow-pot and make it extra beautiful.
4. 1st shot: It's a little hard to tell from the photo. Try to wipe it off with a damp cloth. If it wipes off, its probably just a little debris that got in before the leaf opened, if it doesnt it could be a bit of chemical residue - at any rate, it's not any of the common pests.
2nd shot: These are common marks, can be chemical burn, fungus, even bacteria, but unless it starts appearing all over the plant, not to worry.
P.S. South window is generally considered good light, unless it's shaded by trees, etc. Your Peacelily should be able to make lots of beautiful flowers