Quote: "According to Newton, "Every action has an equal and opposite reaction", which means that to jump up, will push the earth down."
I think that's a very simplistic view, not strictly speaking wrong but it doesn't take into account other factors.
Note: These statements assume you mean Earth as in the planet rather than earth as in soil, ground, dirt etc.
1) The mass of earth compared to a single human being is massively overwhelming.
2) The planet earth is not a perfect conductor, or a single material. It's made up of many substances, some of which transmit kinetic energy better than others.
For example, if you jump up and down on a trampoline then the impact energy is absorbed, stored and even returned with a loss.
The same is true for lose soil, if you jump up and down on loosely pacted earth then it will absorb a great deal if the energy, rather than pass it on.
After all if no energy was ever lost from impacts, and transmitted perfectly all the time then as I hammer on one side of planet earth some one would have to send me a very long distance letter asking me to keep it down they are trying to sleep.
3) The earth has rotational spin, which is a massive kinetic force. Gravity has been holding planets and bay for billions of years, given there mass that's pretty darn impressive even if it is supposed to be the weakest of the four forces (after all a small magnet can overcome gravity very easily).
Therefore you might be able to "push" the immediate soil under your feet down, or perhaps send some vibrations through a building. But you're no more pushing the planet "down" than I am when I jump. Not as a whole anyway.
Quote: "If, to move the earth xxxxxx Kilometer's takes xxxxxx Joules of energy, it follows that xx Joules of energy will move the earth xx Km's, right?
The keyword in Newton's law is equal, if you weigh 1/1000000000000th of the weight of the earth (random amount). Then if you jump 1 meter, the earth will move 1/1000000000000th of a meter, an undetectable amount."
Nope, because I was quoting extreme numbers to overcome extreme forces. Once you overcome all forces with enough kinetic energy, then you can generate some kind of unnatural movement.
Think of me, I used to be a rugby player in school. I'm large, heavy set and I know how to push. (or at least I did in school). Now imagine that it's not only me you are facing but I have a big pile of sandbags in front of me as well, all pressing against a metal sheet. I'm exherting all my power constantly on one side, the metal sheet and sandbags are perfectly absorbing my force.
No matter how many times you run at me with equal strength, or even slightly more strength you can not hope to overcome the two types of material (one is absorbent, the other is rigid) in addition to my own power. This is because to do so you'd have to overcome the properties of all these objects. Their friction, the loss of kinetic energy through the substances and the fact that I'm acting upon them with equal power.
This is more or less what's going on all the time, you'd have to overcome the mass, momentum, and forces acting upon the earth in order to move it down. The most you can do is spread a bit of dirt around and maybe send a very small vibration through the ground for a few miles before it's completely cancelled out.
Quote: "Does your own gravity not pull Earth back the same amount your jump pushed it away?"
No, because if that happened you would have to be extremely dense. You would have to have the same amount of mass as the world. But if that was to happen then earth would "fall in around you" and you'd become more or less the centre of the planet depending on how dense your concentration of matter would be.
Theoretical black holes are a good example, remember the idea is that these are the reminiscence of a collapsed star. Some people may say it has infinite gravity, but that's not true. It has an equivalent amount of gravity for the mass that the star had, but that mass has been super concentrated so you're effectively getting all of that stars energy in a fraction of the size.
In theory earth could orbit a black hole, providing it was compressed down to 4.4mm and orbited with a period of 3cm ( 30mm ). Effectively that would be a very small black hole orbiting a larger one.