Can the curved space-time created by a moving object affect itself? Assume a body with some mass moving in a straight line in empty space without any objects around. Now according to general relativity, it curves space-time around it. Is it possible in any way that the curved space time affect the objects motion?
 A: No. General relativity in flat spacetime reduces to special relativity. If an isolated (massive) object moves with constant momentum in flat spacetime, a reference frame exists in which that object is at rest. Any observable event, like a collapse to a black hole or a radioactive decay, will be agreed upon by observers in all reference frames.
A: Answering the title question: yes, but not in the particular example you provided.
This is an example of a self-interaction, or self-force, and it is, in a well-defined sense, a higher-order correction to geodesic motion. There is a 2009 preprint by R. M. Wald (arXiv: 0907.0412 [gr-qc]) describing the general ideas behind these effects. There is also a 2011 Living Reviews in Relativity (doi: 10.12942/lrr-2011-7) discussing the motion of point-particles in GR. Wald's Advanced Classical Electromagnetism discusses the analogous problem for Electromagnetism in Chap. 10. I refer you to those references (and references therein) for further detail. In here, I'll keep to some conceptual notions.
As pointed out by rob, the particular situation you prescribed is symmetric, so by taking a look at what happens in the particle's reference frame leads you to the conclusion that it can't be "pulled" to any direction. You'll get a Schwarzschild solution (notice that you can't really take a massive point-like particle in GR).
In other situations, however, you can get emission of gravitational radiation by accelerated masses, in a fashion similar to what happens with Electromagnetism (Do accelerating masses generate gravitational waves?). As a consequence, you have a "reaction force" on the body. This means that the geodesic equation (for gravity) or the Lorentz force (for electromagnetism) does not take into account all of the effects involved in the motion of a point-particle and one has some more trouble describing the motion correctly, meaning in particular that the differential equations for the position of the point-particle as a function of time get correction terms, making them deviate from the "usual" solutions.
A: 
Assume a body with some mass moving in a straight line in empty space without any objects around.

A single material object in spacetime generated by it does not move in space but in time. Without other objects around the notion of relative movement in the space makes no sense.
A: The answer to this question depends on the nature of the object and its motion.  The standard Kerr solution to Einstein's equation, which describes a spinning black hole, can be expressed in the form (Kerr-Schild coordinates):
$$g_{ab} = \eta_{ab} + C \ell_{a}\ell_{b}$$
where $\eta$ is ${\rm diag}\left(-1,1,1,1\right)$, which we will henceforth call the "Minkowski metric" and $\ell$ is a null vector relative to both $\eta$ and $g$, and $C$ is a constant.  There  is a particular form that $\ell$ and $C$ have to take, but for the scope of this question it's not important.  What is important is that since you have reduced the metric to a minkowski metric and a null vector on a minkowski background, you can freely Lorentz transform it, and if the base metric describes a non-moving spinning and charged black hole, the boosted metric describes a moving one.  But all of this is just coordinate transforms, and no physics has happened.  This means that, from the perspective of back-reaction on the background, we still have a timelike Killing vector, and the spacetime of a black hole moving at constant velocity is still static, and no back-reaction happens.
But, if you go and extend the solution to describe a system where a piece of matter has a physical extent, or where the black hole is accelerating, the situation changes.  In that case, it requires a lot of fine-tuning to get it so that the higher time derivatives of the quadrupole moment of the matter distribution (or the radiation-related Weyl scalars, for the case of vaccum solutions) are non-zero.  This means that the moving object will emit gravitational waves, and those waves will cost the moving object some energy, and it will feel a force.
