Would time dilation change both the position and direction of a spacecraft as seen from the earth? Given that we can calculate all forces on a moving spacecraft including gravity, would the position and direction in which it is travelling be affected by time dilation ?
 A: 
Given that we can calculate all forces on a moving spacecraft including gravity, would the position and direction in which it is travelling be affected by time dilation?

No, think about what the words time dilation mean. 
To you in your fast moving spacecraft, everything inside the spaceship seems normal, time passes normally, the measurements of the size of your instruments, for example,  agree with the figures you calculated on Earth.
You, looking around inside the ship, will not notice anything different, (looking outside the ship is a different matter, but this will just confuse the issue).
But to an outside observer, who can see both you and the instruments of the ship, things look distorted. He would view your dashboard clock as taking far longer to go from a time of 10.50.05  to a time of 10.50.06 than you do. To you, it takes a second, but depending on your speed relative to him,  it could, for example, take 5 seconds on his clock.
That what time dilation means, there is no absolute  time of 1 second duration that you can agree on, but there are laws, the Lorentz transformations, that allow both of you to reconcile your measurements.
My point is that physical laws are the same everywhere, including the direction your spaceship moves when the main engine is on, or when thrusters move it sideways, but your perception and measurements of these movements will vary depending on how fast you travel. 
A: The answer heavily depends on the direction of relative motion between the two frames.
So, suppose that $v_x$,$v_y$ and $v_z$ are the 3 components of the velocities of the spacecraft with respect to a global stationary observer.
Now, if the observer is moving with a velocity $V_X$,$V_Y$, then $v_x$ and $v_y$ with respect to the observer will be affected by the STR velocity addition rule. Although there is no relative motion with respect to the z- component, $v_z$ will still be affected by time dilation.
Thus now you end up with $v^{'}$ which has different components that $v$ and hence has a different direction with respect to the observer.
However, technically, it is not correct to attribute this to "time-dilation" alone and may be thought of as STR effects as a whole.
