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Yucca is very difficult to eradicate. However, if you wish to get rid of the plant, cut it down as close to the ground as possible and then paint the fresh cut with concentrated Round-Up. Keep an eye on it and if you see any shoots, you'll have to repeat this process. This article will provide additional information: https://www.gardeningknowhow.com/ornamental/foliage/yucca/removing-yucca.htm

Yes, it does sound like the cold has damaged it. It can regrow from the stem or roots if they are still alive. This article will help you determine that:
https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm

Allow the cut to dry out a bit before planting. This will help prevent rot.

You will need to determine if it is still alive. This article will help you with that:
https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm

If it is still alive, it will regrow, though the shape may be different.

Also, this article may help you protect it in the future if it is still alive:
https://www.gardeningknowhow.com/ornamental/foliage/yucca/yucca-cold-weather.htm

Yucca plants may be placed in areas of partial sun but the majority thrive well in full sun.

yes it's fine, you can remove them it won't cause any problems. they should be removed anyway, just cut them off as close to the trunk as possible :)