I was wondering if a moon Is in the center of two planets If the gravity will rip It In half or If a moon can rotate around both planets or if the moon manipulates the planets rotation
I'm afraid most of what you're looking for is impossible.
A moon can orbit a binary planet system by going around both planets, but never by going in-between them.
A "figure-8" orbit is highly unstable because for this to work it would need to be free of any outside interference - no stars, no planets, no other moons - nothing could be anywhere near this system, and even then it would just be a matter of time before the system collapses. Of course, life would be impossible in such a system, without a Sun.
An L1 orbit, which means the moon would lie between the two planets, is highly unstable because of the same reason...any kind of outside influence would cause the balance between the two planets to shift, causing the moon to either crash into one of the planets or begin orbiting one of them separately from the other. (which also might be unstable, depending on the difference in mass between the planets and the orbital distance of the moon)
If you want a moon going around two planets, it can do so by orbiting around both of them and always remaining on the outside.
In the center? Short answer no, long answer yes, but...
Orbiting around both? Yes.
If you have a binary planet, the two planets orbit a barycenter. A sufficient distance away, this is gravitationally equivalent to a single planet of the combined mass. So you can have a moon orbiting the barycenter of a binary planet system.
Another possibility depending on the setup is that the moon can orbit at a Lagrange point between the planets. The Lagrange points behave like little barycenters themselves, and some form stable-ish orbits.
This is the closest thing to what you describe: a moon could technically orbit at the L-1 Lagrange point, but such orbits are unstable. The moon could be in a halo orbit, which allows it to spend some time at the L-1 point, before drifting away. It is conceivable that your moon could be in a periodically stable L-1 orbit where it drifts in, orbits L-1 for a time, then drifts out, only to drift back in and repeat the process later.
If you're willing to put the moon at the L-4 or L-5 points, those orbits can be quite stable. This won't be directly "in between" the planets, but it would be at a vantage point where both planets would be very prominent in the moon's sky.
In any Lagrange orbit case, the mass of the moon needs to be much smaller than the planets in the binary, otherwise it will itself affect where the Lagrange points are and make the orbit much more unstable.
For a moon to orbit two planets and stay between them all the time, it would have to be placed in the L1 Lagrangian point, where the gravity and orbital centripetal forces of the two planets cancel out. Unfortunately, this Lagrangian is unstable; it is unlikely that any moon would stay there for very long.
You can, however, have a stable moon in the L4 or L5 Lagrangian points, which form equilateral triangles with the two planets and rotate with the planets as they rotate around each other.
It is not, I believe, possible to have a moon shift between orbiting one planet or the other in a figure-8 or any other configuration. It will have to either orbit one of them, or orbit both at a distance.
If, however, a moon has an orbit that intersects the L1 Lagrangian point between the planets, it might feasibly on occasion shifts its orbit from one of the planets to the other. It would orbit one planet, linger for a while in L1, and then go on to orbit the same planet OR the other. There might have to be some outside force that nudges the moon from orbiting the first planet to the second, such as another moon around one of the planets, or another planet exerting tidal forces on the system.