The earth is accelerating. When you jump, you increase your velocity relative to the earth. As the earth accelerates, it catches up to you.
Sorry. When you jump off an accellerating object you accelerate with it, at a higher rate. And even if due to other factors, the Earth wouldnt catch up to things the way it does.
No you don't, you are moving with the same velocity as when you jumped off, but you don't have acceleration, because there is no force affecting you at that moment.
If there is no force propelling you, you can't accelerate, F=m*a .
And if this was a clever trick, to trick us into saying that there has to be a force propelling the earth, you can simply tell me what makes objects attract in conventional physics, that is, what's the cause of the gravitational pull.