"Anybody can/could make a mistake" or "Anybody can/could make mistakes". The difference is that "can" is more like you're stating a fact about human nature, while "could" is expressing a hypothetical situation. I hope this is helpful!

Neither. Anybody can make a mistake. Anybody could make a mistake. You really need the "a", otherwise your sentence sounds broken. We typically say, "Anybody can make a mistake", but using "could" is still perfectly valid.