Reveal this card before you search. If you reveal a vampire encounter or “Hoax,” discard both this card and that card.
If the hideout or lair searched in had both a vampire encounter and a Hoax (or for some reason multiple vampire encounters and/or Hoax encounters) would Stake remove all such encounter cards or can it only remove one?
Just one. "a" and "that" are singular. When searching the hunters choose the order of resolution, so in the case of a hoax and a vampire, you can choose to discard the vampire.