If your question is what the fiqhi ruling is on cat hair, we have to say that amongst all animals, only dogs and pigs are najis.[1] Therefore, a live cat and its hair are pak. Of course, its excretions are najis[2] and this ruling doesn’t only belong to cats, but the excretion of all haram meated animals and animals that have gushful blood (when their throat is slit) is najis[3], the cat being one of them. Also, in the case of the cat being dead, its hair isn't najis although its body is, because if falls under the category of mitah, and the ruling of mitah or dead bodies is that all of the parts that contain life are najis after death, while other body parts such as the hair that don’t contain life aren't.[4]

As for in prayer, one’s clothing mustn’t carry any parts of the body of haram meated animals (regardless of whether it is of the ones that contain life or not), therefore, since cats fall under this category, according to most marja’s, if their hair is carried by a person praying, it will invalidate the prayer.[5]

It is only Ayatullah Sistani who says that the clothing of a person praying mustn’t have the excrements, sweat, milk or hairs of haram meat animals, but if only one hair for instance is stuck to one’s clothing during prayer, it won't be problematic. The same goes for a person whose clothing doesn’t have any of the such, and the person is only carrying them with him, like in a box or container.[6]

As for body moisture of haram meated animals, all marja’s say:

If the saliva, nasal mucus or any other moisture from haram meated animals, such as cats, are on the body or clothing of a person praying, if it is still wet, the prayer is void, and if it has dried and there is no longer any substance left, the prayer is correct.[7]

As for hygiene and health, it is said that cat hair brings infertility for women, but this is exceeds our expertise and doesn’t concern us and doctors are the ones to be referred in this regard.[8]