Some mysteries will require you to place eldritch tokens on various spaces. When such a mystery is solved, do you remove the eldritch tokens placed by that mystery? I have concluded that the tokens are meant to be used generically across cards and effects that cause them to appear, so my thinking is that the answer is "no, they remain on the board, even if they currently have no meaning."
According to the Reference guide, page 8:
When the active Mystery is solved or removed from play, discard all tokens on it or placed by it. Then draw a new Mystery, placing it on top of any solved Mysteries.
So, in almost all cases, tokens placed my mysteries would be removed, and then replaced if the next drawn Mystery card requires additional tokens to be placed on the board.
If an individual Mystery card specifies differently, the text on the card supersedes the text in the rules. E.g., there is a Cthulhu-specific mystery that specifies that all eldritch tokens placed as a result of that card remain even after the mystery is solved.
Yes, you remove all the token as soon as the mystery is solved. They serve no purpose afterwards. They are meant to be used generically iff there is a card that require them.