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Once the flowers have faded, you can trim the entire stem back to the bulb with sharp scissors.

This article will help with getting them to rebloom: https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-bloom-again.htm

This article should help: https://www.gardeningknowhow.com/composting/ingredients/eggshells-in-the-garden.htm

I would try moving it to somewhere with less direct light, at least until the leaves have grown a bit. It sounds like the leaves grew some in darkness and became sunscalded when placed in direct light. You can kind of think of it as a sunburn for plants. If the leaves are not use to the light, they will get scalded when placed in light. In the future, just gradually move it into direct light so the leaves can get use to going from the dark to the full light.

Yes, as long as you make sure some roots come away with the pup.

Amaryllis bulbs are typically shipped with a bud or two ready to grow so that you get a bloom almost immediately. But sometimes the grower mistakenly packs a bulb that is not budding or the bud simply gets damaged in shipping and does not grow. You can keep them for the year and they will bloom in the spring. Here are directions on doing that: https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-care-instructions-how-to-care-for-an-amaryllis.htm