First of all, after the correct comments, you are asking (or mentioning) about two main things:
why is one twin getting older. since this is not your main question, I will not go into this
why can the twin on the spaceship see things move faster then light on earth.
I believe your main question is 2., why does the twin in the spaceship see things move faster than c on Earth, an why is SR violated. You can already feel that SR is not the right thing for this phenomenon. That is in part why GR was developed, GR describes gravitational time dilation, and that is what to our current knowledge best describes the events you are asking about. Basically in short version, SR states that nothing can move faster then c, when measured locally in vacuum. Your measurement is not local. You are trying to measure speed from far away (from a different gravitational zone).
If you try to explain this with SR, you will realize that both twins could symmetrically say that the other's clock moves faster. Which one is right? Constant speed is symmetrically relative. You need acceleration. Acceleration is absolute. That is why you need GR. The problem with the SR explanation is too, that it disregards the period of turnaround, and how it works in real life. Turning around with a spaceship needs time and energy. It needs acceleration, and that is equal to a gravitational zone according to the equivalence principle. That is the specific period during which the traveling twin will see the other's clock tick faster, and the twin on Earth will see the other's clock tick slower. SR cannot give you that with a real life explanation (or at least it is much more complicated).
You are on the right track, just need a few things to clarify:
you are correct, this is due to gravitational time dilation (that is where you ask about the "during acceleration" part of the journey)
because of the equivalence principle, the accelerating ship has the same effect on the clock as a gravitational zone (slowing down the clock relatively)
light travels at speed c in vacuum when measured locally
you are right, you can measure a different speed then c when you measure the speed of light from a far away observer's view
You can measure a speed different from c even when measured locally in a medium (not vacuum), but your case is about the nonlocal measurement.
The main thing is, that you need to measure the speed of light from a far away observer's view, and the observer needs to be in a region of space that has a different stress-energy (gravitational zone) then where the light is actually traveling.
There are two main cases:
Let's say that you are measuring the speed of light as it passes next to the Sun, and you are measuring it from Earth. This is the Shapiro delay, and you will measure a speed less then c, because of the Sun's stronger (relative) stress-energy, and Earth's weaker stress-energy (relative to the Sun). The clock on earth ticks faster, and you divide the path of light with a time that is more (relatively), so you get a smaller speed. The path is longer too, because of gravity, but let's disregard that.
And yes, contrary to popular belief, it is possible to measure a speed more then c when measured from far away. When we say the maximum speed is c, we mean when measured locally in vacuum. If you would measure the speed of light as it passes next to Earth, and the observer is at the Sun, you would measure a speed more then c. How is it possible? It is because your clock at the Sun ticks slower, and you will divide the path of light with a smaller amount of time (relatively), so you get a speed more then c. The path is a little bit longer because of gravity, but let's disregard that (actually the time component is more dominant so the path length won't matter in this case).
So basically yes, the answer to your question is if you look from the spaceship as it accelerates, you could see things move faster then c on Earth. This is because you are measuring speed non-locally, from far away. SR only states that if you measure speeds locally, then you cannot measure speeds faster then c (in vacuum). So SR is not violated.
Please see here why you would prefer GR to explain this:
How is the classical twin paradox resolved?