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Yes, the plant is getting ready for flowering, which will either be white or cream colored. Once it's finished, it will begin forming brown seed pods, which you can collect and plant or simply remove the faded flowering stalk and discard.

Either time is fine. You should start it in a container though. This article will help: https://www.gardeningknowhow.com/ornamental/foliage/yucca/propagation-of-yucca-plant.htm

In terms of foundation, you are fine. Their roots tend to be shallow. As far as pipes, as long as it gets enough water (which it typically needs little of) it should not go after the pipes.

This is the flower, which you can remove if you want.

Sounds like it may be suffering a little transplant shock. This article should help with that: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

They prefer well-draining potting soil. This article will help with repotting: https://www.gardeningknowhow.com/ornamental/foliage/yucca/separating-and-repotting-yucca-offshoot-pups.htm