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There are several things that can make the peace lily leaves turn yellow -- letting the soil stay too damp damages the roots so they can't transport water, and the leaves will turn yellow; letting the soil stay too dry, thus not containing any water for the roots to transport, will turn the leaves yellow; too little light will cause leaves to turn yellow. To decide what the problem is, test the soil in the bottom of the pot with a long skewer or moisture meter. The soil should read (or feel) only slightly damp before you water again. When you water, water enough that you get 1/2" run off, and don't worry about dumping it out. If soil moisture seems okay, move the plant so it gets more light. If you can read easily in the place where you put the plant, there should be enough light for it.

Yes, you can remove the black leaves. Cut the leaf stems off at the bottom, where they emerge from the rhizome, which is under the soil. The brown things on the stems are scale insects, and the sticky sap is called honeydew, and is produced by the scale. You can get rid of them by using the techniques in this article: https://www.gardeningknowhow.com/plant-problems/pests/scale-bug-how-to-control-plant-scale.htm

First, how big are the brown tips, and how many? If there's only a few tiny (1/4" or less), that's just ordinary usage. If there are bigger tips (1/2" or more), and on most of the leaves, that's a sign that the soil is too wet, and the roots are in distress. You can easily trim the tips off - use scissors, and trim so that the natural shape of the leaf is preserved. However, if you don't allow the soil to dry more between waterings, you will eventually lose the plant. Test the soil in the bottom of the pot - it should be damp, but not too moist. (Of course, the other side of the coin is that you don't want the soil to get too dry. Some people routinely let peace lilies start to wilt before they water them, but I don't think that's a good practice either.)

The roots can talk to you through the leaf tips after you trim them off - if they're still too wet, the tips will come back, but if the soil has dried out enough, you won't see any more tips.

Peace lily leaves tend to turn brown when the soil is too wet, because the roots rot. When repotting, you need to use a free-draining mix; a cactus soil mixed 1/2 & 1/2 with perlite would be very good. You might want to go ahead and repot your peace lily in some new soil. Also, make sure it dries sufficiently between watering. Check the moisture well down into the pot, not just on the top. An electronic moisture meter would help. Here are a couple of articles that could help you: https://www.gardeningknowhow.com/houseplants/peace-lily/peace-lily-plants.htm
https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm

Cut off the shriveled and damaged leaves, set it in a bright spot (out to the cold, obviously) and test the soil for moisture level near the bottom of the pot, making sure the soil does not stay too wet - it should be maintained at a 'slightly damp' stage. If the roots are viable, they will send up new leaves. https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm