Can a goalkeeper use his hands in this situation?

If the intention in the defender's mind was to pass the ball to you, then you can't pick it up. If he tries to clear the ball and mishits it, then he is clearly not intending to pass it back to you, so you can pick it up. Whether you take it off the toes of a striker or not is irrelevant. If the defender didn't intend to pass it, you can pick it up.

the goalie can pick up the ball with his hands only if it is played last by either the other team, or by his own team via a header or chest pass. (thus, if his own team kicks it to him and he picks it up, it's a hand ball on him, and since it's in the box, it's a penalty kick) FIFA changed the rules to make it harder for defenses and open up the game more, because it's too easy for a defense to kick the ball back to the keeper from almost anywhere on the field, so this way they can't do it unless the defense heads it back.

I've never had that happen.

It is still technically a pass back because he kicked it.

So no hands allowed.

If there is an imminent threat you still gotta use your feet. Boot it out of bounds if you have to.

Both Yes

It has to be an intentional backpass... i have seen several incidents of these in the EPL and when i play locally.