Does a moving object curve space-time as its velocity increases?
Start with the gravitational field of the Sun. We are effectively stationary with respect to the Sun, because our relative speed is much less than $c$, and the Sun is rotating at well below relativistic speeds so we expect its gravitational field to be well described by the Schwarzschild metric. And indeed this is true: Newton's law of gravity is the non-relativistic limit of the Schwarzschild metric.
The metric tensor is invariant with respect to coordinate transformations, so if we take some observer moving at near light speed they would also find the gravity round the Sun to be described by the Schwarzschild metric. It will not look the same in the observer's coordinates, that is the individual components $g_{ij}$ will be different, but it will be the same tensor. Since in the observer's frame they are stationary and the Sun is moving, the conclusion is that velocity does not change the spacetime curvature.
Incidentally, this is why a fast moving object does not turn into a black hole.
I assume you're asking whether a moving objects curves spacetime differently than a stationary one as its velocity increases.
Strictly speaking, no: it's the same spacetime geometry either way. The spaceship warps spacetime either way, and all we'd be is talking about it in a different frame. Because of this difference of frames, in a sense the gravitational field is different even though the geometry is the same.
If this spaceship, according to special relativity, gains mass as a factor of y as it approaches c, then its gravitational field should increase in strength as well.
This is not quite right. First, the spaceship does not gain mass. It's true that quantity $\gamma m$ is sometimes called relativistic mass, but this term is redundant with energy, bad at its intended purpose of preserving a superficial resemblance to Newtonian mechanics, and depreciated in physics. In special relativity, mass is invariant: $(mc^2)^2 = E^2 - (pc)^2$ is the same in all inertial frames.
Which is just as well, since the 'gravitational charge' isn't mass, but energy. But it is not a simple proportional increase when we view the spaceship's gravitational field in a frame in which it has a lot of energy.
This shouldn't be surprising if you know a bit of electromagnetism. A moving charge produces an electromagnetic field that has both electric and magnetic parts, since the motion of the charge, i.e. the current, matters. The electric field is enhanced in directions perpendicular to the direction of motion, which we can picture as the initially spherically symmetric field lines getting Lorentz-contracted, thus 'squishing' them close together in the perpendicular directions.
The gravitational analogue of electric current would be momentum, but because gravity is spin-2, stress in addition to energy density and momentum density is relevant to how spacetime is bent. You can see this described in the stress-energy tensor. So the gravitational field is more complicated, but it has an analogous behavior of being strengthened perpendicular to the direction of motion, although its quantitative behavior is different.
In the limit of lightspeed, the electromagnetic field of an electric charge turns into an impulsive plane wave, and the gravitational field of a point-mass behaves analogously, turning into a vacuum impulsive gravitational (pp-)wave.
Looking at the (spatial) velocity of an object alone is like looking at a plane and considering only its north-south velocity and neglecting its east-west motion.
Timelike objects are like cars with only one gear, no brakes, and a gas pedal that is stuck in place. They continue to go forward in time no matter what; all they can change are their direction through spacetime--how much velocity is going forward in time vs. how much is used to travel distances in space. (The steering wheel still works, but that's all, so to speak.)
When you consider both of these velocities together as one quantity, you find that this "four-velocity" of a given object has constant magnitude: the speed of light itself.
Similarly, the "four-momentum" has constant magnitude, proportional to the rest mass of the object.
These are all results from special relativity. General relativity is a bit more nuanced (not in the sense that four-velocity and four-momentum no longer have constant magnitudes--they do--but in how those magnitudes are calculated using the metric).
While different observers will disagree on obviously frame-dependent things like spatial velocity, they will all agree on the magnitudes of vectors in this fashion. You've been told that gravity depends on mass. I'm telling you that it depends on rest mass--which all observers agree on in this way.
Because spacetime boosts are analogous to rotations, boosting a particle and looking at its gravity just results in all spacetime variables (like the metric) being boosted as well. These transformation laws can be tricky for some tensors, but the underlying physical system can still be understood as equivalent to that of a stationary particle. It's just being looked at through another observer's eyes.