So hypothetically if we could kick-start the core of the planet Mars and "turn on" its magnetic field (by say inducing an asteroid moon-ring for lack of a better description) thus shielding its atmosphere from the Sun and begin "terraforming it. My question is would the magnetic field speed up the rotation of Mars and would that pull Mars in a bit closer inside the habitable zone of its orbit around our Sun thus warming the planet even more and increasing the terraforming process? Would this become a danger to our own world?
No, no and no. Though you will get a stronger magnetic field with increased rotation, if you increased the magnetic field through other means it would not increase the rotation rate. An active magnetic field would have no effect on the orbit of Mars, in fact you could say the larger magnetic field would create a slightly great 'push' from the solar wind on Mars. But as that effect is extremely small in comparison to the planets mass and orbit, it would not likely even be measureable. It definitely not would pose a danger to Earth.
Also this question has been asked before within the stack exchange so browsing some of those should help you as well.