The rule book under Progression of Evil a.) Drawing a Black card (page 10-11)
Yes, you follow the instructions on the card Special Black cards or Standard black card or place the card in its respective quest, regardless of whether any Knight has begun a quest. The instructions are on the card, or detailed on the pages listed here. Excalibur cards move the sword towards the evil side of the river. Grail cards like Despair can fill up the quest before you even start, and if it does you lose the Grail Quest and turn it over, further Black Grail cards place siege engines outside Camelot.
The very end of section a). explains that a quest immediately ends when you place the final black card (or Saxson/Pict) to fill a particular Quest, and tells you to consult the particular quest for win/loss conditions. In general, you will add a number or White(win)/Black(lose) Swords to the round table, add a number of Siege Engines/Draw White Cards, and whether the particular quest is flipped over or not. This information is usually printer on the Quest cardboard itself, but is explained in detail within the rule book.