What is the ruling on covering a woman's bier with a black cloth, and does this have a basis in Sharia, or is it an innovation (bid'ah)? And what is the ruling on covering the grave when burying a woman, and is there sound evidence for this, or is it a reprehensible innovation?

It is recommended to cover a woman's bier with a more concealing garment, like the dome placed over the bier of Fatimah (may Allah be pleased with her). This covering is not required for a man. The most deserving people to place a woman in her grave are her mahrams, then women, because it was permissible for them to see her during her lifetime.