Space curvature and its rotation As I learn, the gravity is caused by the spacetime curvature. Is there any gravitational/magnetic field that really exists around the Earth, or any other planets?

Since the planets/stars rotate, does this cause the space time curvature to twist in the direction of their rotation? If it does then the gravitational pull should also twist the the path of the objects falling(though we may not perceive the curvature of this path, we should be 
able to calculate at-least theoretically)
Unlike gravitation, which is entirely a result of space-time curvature, magnetic fields do exist and there is a magnetic field around the Earth and many other planets. 

(There is a deeper level of understanding of electromagnetism called Quantum Electrodynamics which explains how fields work in terms of more fundamental concepts, but unlike gravity, the E-M field does exist. (And there is hope -- so far unrealized -- that eventually all of the fields of nature plus gravity will be explained as aspects of one yet deeper thing.  But not today.))

Frame dragging -- the distortion of space-time by a rotating body -- does exist and the link suggested by @PM (en.wikipedia.org/wiki/Frame-dragging) is a good place to start.  Frame dragging has been measured by careful observation of satellite orbits, and possibly by astronomical observations.  The Gravity Probe B satellite measure a closely related effect.  As far as I know, the effect is too small to have been measured in the lab.

Frame dragging does indeed affect the path of a falling body, though for the Earth the effect is really, really small.  It would be large near a millisecond pulsar or a rapidly rotating black hole.  It's also large for merging black holes and is an essential part of the calculations of the gravitational waves emitted during the final rotations before merger.

The effect can be precisely calculated and agrees perfectly with the (still rather crude, alas!) experiments and observations.