Spencer didn't ask for the tiara back from his sister. The Spencers keep their tiaras in a vault in London. Any time Diana wanted to wear it, she had to ask the head of the family if she could use it, and the head of the family had to authorise it's removal from the vault (for insurance purposes). It didn't remain with Diana. It remained (and remains) in the vault for security.
Diana received an amethyst demi-parure of a necklace, bracelet and earrings as a wedding present. I never read who gave them to her, but they were amethysts and they were a wedding present.
The suite from the Sultan of Oman are sapphires, not jet. They were listed as sapphires in the Morton Windsor Wealth book and she wore them with her blue gowns (though because they are so dark, they didn't seem to show up well).
Like all of Diana's other wedding presents (that were gifts to HRH The Princess of Wales), they went back into the royal family's vault after her death so that William and Harry could have them for their wives. Diana had a number of pieces that were gifts to her as a person rather than her as PoW, and these pieces have either gone into the vault for safekeeping for her sons (the majority of the pieces), to her sons directly, or been sold (Swan Lake suite).
Diana's jewellery was estimated (by a professional), based on the worth of the stones alone, not their royal provenance, to be worth about L2million in Andrew Morton's mid-1980s book about the Windsor wealth. Nowhere near $40 million and that book came out after Diana had received her best pieces. A lot of the gems Diana received from the Windsors were older pieces that would have to be recut significantly to get the sparkle modern jewellery lovers prefer, but that would require a loss of carat-weight, thereby decreasing the worth of the piece.