Is it permissible for my sister to keep the elderly dog out of care for it? Will she be sinful for that? And is the name "Angelic Care" suitable for a childcare facility, and does it not resemble Christians?

It is forbidden to keep dogs without need, such as for hunting or guarding livestock and crops. It is permissible to keep them for guarding houses by analogy, when there is a need. Whoever keeps a dog for reasons other than these will have a qirat or two qirats deducted from his reward daily, and angels do not enter a house in which there is a dog. As for naming a nursery "Angelic Care," it is not appropriate, due to the suspicion of imitating disbelievers in describing angels as females, and imitating Christians, and because it is an exaggeration or a lie.