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Yucca is a shrub-like plant (occasionally found in tree form) with sharp, sword-shaped leaves. It is popular as both an indoor and outdoor plant. For more information on yucca plant, this article and list of other resources should help: https://www.gardeningknowhow.com/popluar-houseplants/yucca.htm

I would wait until spring. The plant should survive, but you won't know the extent of the damage until spring, so it will be best to wait until then to prune so that you can better judge exactly what needs to be taken off.

This is normally cause by either overwater or underwatering. With yuccas, people tend to overwater them. Make sure you are only watering when the soil is dry to the touch.

There is a chance that the trunk was damaged and this is why it is not able to support itself. It may or may not make it. Only time will tell. Put a splint on the trunk and leave that in place through spring and summer. If it is able to heal itself, it will do so in the spring and summer. If it is not able to heal, it will either die or will need a splint forever in order to stay upright.

It sounds like they are suffering from transplant shock. This article will help you:
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm