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Philodendron scandens, heartleaf philodendron is easy to propagate from stem cuttings. Make a 3 inch cut just below a leaf, remove all but the top leaf and look for a leaf node. Place the cutting in a small glass of water with the node submerged. (The node is where new roots will emerge.) Place the glass in medium light - no harsh, direst sun. When the roots are 1.5-2 inches long, pot the new plant in loose potting soil. Voila!

You can do this as long as you do not damage any portion of the plant while doing so. I do recommend leaving it to climb, but it would, certainly, survive trailing.

No! This is part of its natural cycle. Just cover these with more soil is they are unsightly. Otherwise, I would leave it as it is.

The best thing you can do is bring it back to its proper temperature, and water a little less. Cut off any dead growth, and with a little luck, you should see new sprouts eventually.

Unfortunately this can be caused by several different things. Over, or under watering could both do this, as well as infection. Can you include a photo? This can help to identify what is going on. Here is our collection of articles that will help: https://www.gardeningknowhow.com/houseplants/philodendron/

I would say this will be fine, but you will want to make sure you leave some leaves so that the new shoots have some way to generate energy. You won't want to cut off more than a third of the plant at once anyways.