1) It's slang. I spell it "flaky," without an "e," but I think "flakey" is also common. 2) I don't think "flaky" means "passive-aggressive." I think it means "unreliable" or "strange, eccentric." 3) "Passive-aggressive" means expressing anger in a sneaky way. Suppose Mary has a dog, John marries Mary, and John thinks of it as "Mary's dog." Mary asks John to go buy dog food. John doesn't want to do it. An aggressive response by John would be: "I won't do it. It's YOUR dog, YOU buy the dog food." A passive-aggressive response would be to say "Certainly, dear," go out and buy fifty bags of dog food. When Mary says "Oh, no! Why did you do that? I don't have any place to put it, and some of it will spoil before it all gets used." John replies "Well, you told me you wanted to buy dog food, and I bought dog food, didn't I?" The CORRECT response is for John to say "Certainly dear, how much and what kind?" and return with the right amount of the right kind.

I agree. I'd understand a 'flaky' as meaning that that the person is slightly unstable. A flaky person may behave in a strange or unreliable way.

"Flakey/flaky" means unpredictable, or non-committal. It's a stretch of the imagination to say it means "passive-aggressive".