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Spring and Autumn will be the best times. You will have to look after the plant carefully in order to prevent transplant shock. This article will help: https://www.gardeningknowhow.com/ornamental/foliage/yucca/transplanting-yucca-in-garden.htm

That is a Yucca plant! This article will help: https://www.gardeningknowhow.com/ornamental/foliage/yucca/caring-for-and-landscaping-with-yuccas-outdoors.htm

According to the University of California, yucca plant is safe around pets and humans as far as ingesting plant parts.

Some varieties have sharp spikes so you may want to avoid those. Other than that, site your yuccas in full sun and dry areas.

Here is information on varieties of yucca:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/yucca-plant-varieties.htm

Cactus soil is thinking in the right direction, though they don't really like acidic soil. You will have to "sweeten" it with dolomitic lime, and it should flourish.

This article will give you a better idea of what kind of soil these plants like: https://www.gardeningknowhow.com/ornamental/foliage/yucca/soil-for-yucca-plants.htm

This is a legitimate concern. There are many species of yucca, some slim and delicate compared to the larger varieties which can develop massive, expansive root crowns at the base. I'm not sure which your neighbor has planted, but if its a large growing species, then your boundary wall may be in for trouble.

It sounds like you have 'pups'.
This article will help you.

https://www.gardeningknowhow.com/ornamental/foliage/yucca/separating-and-repotting-yucca-offshoot-pups.htm