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As long as it is in a container, this will not be too difficult. Here is a detailed article on their care: https://www.gardeningknowhow.com/ornamental/cacti-succulents/christmas-cactus/advice-for-christmas-cactus-care.htm

Yes, but it will need some recovery from this blooming season. These have specific requirements before they will re-bloom. This article will give you more detail about this process: https://www.gardeningknowhow.com/ornamental/cacti-succulents/christmas-cactus/how-to-make-a-christmas-cactus-bloom.htm

If you could include a photo this would help me to identify what you are referring to. Now, about flowering... This plant requires a bit of a special set of requirements in order to bloom. This article will explain how to get them to re-bloom: https://www.gardeningknowhow.com/ornamental/cacti-succulents/christmas-cactus/how-to-make-a-christmas-cactus-bloom.htm

Purple coloration is usually due to onw of two things. This can be caused by cooler temperatures, or the more likely culprit... Lack of phosphorus. It is recommended to feed these occasionally.

This article will give you more information: https://www.gardeningknowhow.com/ornamental/cacti-succulents/christmas-cactus/advice-for-christmas-cactus-care.htm

It could be partially responsible for the poor growth. I would say an egg was left in the soil, and hatched and grew there. Interesting how species that don't belong get so far away from where they came from.

Yes, these are one in the same. Now there is an easter cactus also. The only difference between them is the length of light it takes to flower. Here is an article for the care of this: https://www.gardeningknowhow.com/ornamental/cacti-succulents/christmas-cactus/advice-for-christmas-cactus-care.htm