Effect of gravity at near-lightspeeds Let's say I'm in a space station, hurtling towards our galaxy nearly close to the speed of light. From my reference frame, I see the galaxy coming towards my ship at the same speed.
I pass the Sun, and am affected by its gravity.
From Earth's point of view, the gravity of the sun deflected my spaceship's trajectory.  From my point of view in the spaceship, the trajectory of the entire galaxy changed very rapidly.
From my reference frame, how did my space station cause an entire galaxy to change course?
Originally had multiple questions, removed all but one
 A: That is why physics was invented :).
Lets take it a step at a time:

Let's say I'm in a space station, hurtling towards our galaxy nearly close to the speed of light. From my reference frame, I see the galaxy coming towards my ship at the same speed.

Fair enough. True also in Newtonian mechanics, there is no acceleration and it is just a matter of coordinate choice for the rest frame. There is no cause and effect.

I pass the Sun, and am affected by its gravity.

Now acceleration is introduced, and this means there is a central force as far as Newtonian mechanics goes. For General Relativity, the space is distorted and the space station follows the geodesic. The cause of the change is the force/geodesic.

From Earth's point of view, the gravity of the sun deflected my spaceship's trajectory. From my point of view in the spaceship, the trajectory of the entire galaxy changed very rapidly.
From my reference frame, how did my space station cause an entire galaxy to change course?

That is why we have developed physics. The coordinate system change one makes by "earth's pov"  and "my pov" is not causal. The cause of the changes comes from the forces in the system or  in GR the space distortion the Sun's gravitational field causes in its area of measurable influence. How the coordinate system is described mathematically is only a matter of convenience. You could take the pov of an ant moving on the earth. You would need complicated geometric calculations to see why the spaceship is changing rapidly while going for food, but the relation is mathematical, not causal.
The epicycle geocentric view comes to mind. It is a correct mathematically coordinate system but has little correlation with the forces that become apparent in the heliocentric system. So your pov is a spaceship-centric pov.
A: This is sort of the same as Anna's answer, but I'd like to put a slightly different spin on it.
As Anna points out, there are two different co-ordinate systems involved: one for the observer sitting on Earth and one for an observer in the freely falling spaceship, and the situation looks very different for the two observers.
Each observer can (in principle) measure the stress energy tensor then solve the Einstein equation to give the curvature tensor. The key thing to note is that these tensors are co-ordinate independent i.e. both observers will calculate the same stress energy and curvature tensors.
However, although the tensors are co-ordinate independent the representations of them in the two co-ordinate systems will be different. We normally write the tensors as a 4 x 4 matrix, and the two different observers will calculate different values for the elements in the matrices because they're using different bases.
So it's not correct for the observer in the spaceship to think the galaxy is somehow being deflected by his gravity. Actually strictly speaking it's also incorrect for the observer on earth to think the spaceship is being deflected by the Sun's gravity. The gravity, i.e. the curvature, is not attached to any particular body. The solar system and the spaceship together (and in principle the rest of the universe) produce a curvature then both of them move in response to that curvature. The difference seen by the observers is purely down to them using different bases to represent the tensors.
