You have correctly determined that there is no upper bound on proper velocity.
In the rest frame of the Earth and the distant star, time is measured with coordinate time $t$. For very slow observers, their “proper time” (the amount of time they experience, related to the parameterization of their worldline in 4D space) represented by $\tau$ is approximately equal to coordinate time, $\tau\approx t$, so there’s no time dilation.
There is no physical reason why proper velocity $u^x=\text dx/\text d\tau$ (the velocity measured by observers onboard the ship) can’t be arbitrarily high: the observer on the spaceship can see the journey from Earth to the star take an arbitrarily-short period of time. However, for fast speeds, proper time and coordinate time become different. The Lorentz factor, $\gamma=\text dt/\text d\tau$, goes to infinity as coordinate velocity $v^x=\text dx/\text dt$ goes to the speed of light $c$, so chain-ruling, you get a finite maximum value of coordinate velocity $v^x$ even when proper velocity $u^x$ becomes infinitely large. There is also a geometric explanation that can be given as to why you can’t have coordinate velocity vectors with magnitudes greater than $c$: given the Minkowski metric (in 1+1 dimensions),
$$\eta_{\mu\nu}=\operatorname{diag}(c^2,-1)$$
and for a timelike vector having $\eta_{\mu\nu}u^\mu u^\nu=\eta_{\mu\nu}v^\mu v^\nu\gamma^2=c^2$, you clearly arrive at $v^2<c^2$ for the vector to stay timelike (e.g. not be lightlike or go back into its own past).
That is, an observer on Earth will always see the ship travelling slower-than-light, and will see them arrive correspondingly and their clocks ticking slower, while the observers on the ship may not agree and may perceive a shorter time. Here the twins paradox is resolved by the accelerations that cause the ship to start/stop at Earth/the distant star.
It may be argued that, while the spacecraft was moving, the space ahead of it underwent relativistic contraction, according to the observer in the spacecraft, so that the spacecraft travelled a distance less than ten light years.
Yes, this is precisely what happens.
The fact remains that, when the observer reaches the star, and stops, and measures the distance to Earth, that distance will be found to be ten light years.
Only if they decelerate. If they turn around without slowing down, they’ll still see a less-than-10-light-year distance back to Earth that they could have travelled at sublight speeds. But if they change velocities, then they are no longer sitting in an inertial reference frame, and basic special relativity stops applying immediately until they stop accelerating.
Therefore, according to the observer on the spacecraft, the spacecraft has travelled ten light years in 7.5 years. It has travelled faster than light.
Also wrong. In their own frame, the observers on the ship will see light still travelling at $c$ in any direction relative to them. This is the key part of relativity: $c$ is an invariant speed, and no matter what you do, you’ll always see yourself going slower than light (and nearby photons going by at exactly $c$) in any inertial frame.
