If I push or hit an object in space will it rotate or move along a straight line? If I push or hit an object in space (vacuum and no gravitation) in direction what is not going trough its centroid, will it rotate or move along in straight line? 
I expect that on earth it will depend on what is less difficult for the object (rotation or linear movement). So the object will do some kind of combination of both movements (rotating and also moving along the direction of impulse or force).
But how could an object "decide" what to do in space, where is not resistance?
 A: If you give a tangential force it would rotate. If you give a force at centroid, it will move in straight line. Along any other point, between tangent and centroid , it will show joint motion.
Splitting of force into tangential and along centroid will also depend on shape of object. Like you cannot give a pure rotation force to a rod which is not pivoted at any point and pivot is its momentary resting inertia. So it will both rotate and move forward even with just tangential force which will have a linear component as well. More is the mass concentrated at centre of mass more will rotaional component.
A: If the line of action of the force is is not through the centre of mass you can transform the original force $\vec a$ by adding two forces $\vec a$ and $-\vec a$ (net force zero) acting at the centre of mass as shown below:

You can now consider these three forces as follows:
a force equal in magnitude and direction to the original force but passing through the centre of mass.
This force will produce a translational acceleration of the centre of mass of the object.
and
a pair forces equal in magnitude to the original force with one in the same direction as the original force and the the other one in the opposite direction.
This pair of forces is called a couple and produce a torque of $ax$ which is independent of any axis of rotation that you may choose.
A couple only produces rotational acceleration.
If the original force is through the centre of mass no couple is produced and so the object does not undergo rotational acceleration.
A: Any linear force not going through the centre of mass will create torque, which I hope you know, is related to how far from the centre of mass the line of force is.
So, if you manage to hit the object exactly at its centre of mass, i.e. the line of force is directly passing through the centre of mass, then it will show NO ROTATION. It will go straight ahead in a line.
But, if you fail to do so, i.e. the line of force misses the centre of mass, it will show BOTH kind of motions, Rotational and Linear. It will go straight ahead in a line as in previous case, but will also rotate. How much is the speed of rotation depends on how badly you missed the centre of mass.
But in both cases, the total momentum will be (has to be) same.
A: If not influenced by any other forces then after pushing It will move in a straight line and most likely rotating as it goes. It would be real hard to push it without giving it some kind of rotation but it will always move in a straight line.
A: If the net hitting force passes through the center of mass of the object being hit, it will have linear movement, otherwise, it will have some rotation as well as movement.
A: If the line of action of the applied force is not through the "centroid" of the body (more appropriately the centre of mass, shortened as CM), then the body will both rotate and translate linearly.
What does this mean? Well, consider a wheel rolling on the road. If you focus your attention on the centre of the wheel (which, coincidentally, is also the CM of the wheel), you will see that it moves in a straight line. Any other particle on the wheel moves in complicated paths. This is the speciality of the centre of mass of a system, which moves in a straight line path in absence of a net external force on the system.
So now imagine you drive in a car alongside the wheel, and you adjust your speed so that the centre of the wheel appears to be stationary (just don't crash your car. It's precious). What you essentially did was move into a frame of reference having the same velocity as the wheel. Now if you look at the wheel, the wheel seems to be rotating about its centre! Whoa!
So you use this knowledge, and when you stop the car, you say that the wheel is both, translating (it's CM moves at a constant velocity), and rotating (in the frame of CM, the wheel is rotating). In fact, the above discussion applies to all bodies.
Now, when you apply a force not passing through the CM of the body, it has a torque about the CM of the body, and thus the force plays two roles. It not only accelerates the centre of mass, but it also gives angular acceleration to the body, that is, if you stick to the frame of centre of mass (here the frame is not inertial!), you will see that the body is rotating about the CM, but its angular velocity (rotational speed) is increasing, thus it has angular acceleration. All this ultimately means the body both translates, and rotates about the CM (although the "rotation" can be only seen properly in the frame of CM).
