The one thing to keep in mind is that in order to perform a gravity-assist maneuver, you need to be able to enter a hyperbolic orbit around a given body that is moving relative to your destination. And, in order to be in such an orbit, there is a specific range of velocities for every object that you must have (dependent on mass of the object). So the fastest you can get to by gravity-assist is much less than relativistic speeds because at relativistic speeds, you would not be able to enter into a proper hyperbolic orbit.
It is true that at any high speed, a flyby constitutes a hyperbolic orbit; however, to use a gravitational slingshot, you need to enter against the object's motion and exit with the motion from that object's point of view. At relativistic speeds and for most regular bodies, your orbit would closely resemble a straight line, there could be no gain of velocity.
A good gravity assist works if you can ensure that your hyperbolic trajectory minimizes the angle $\theta$ between the assisting body's trajectory and the spacecraft's exit trajectory. It is given by:
$$\theta=cos^{-1}(1/e)$$
Where $e$ is the eccentricity of the orbit and must satisfy $e\geq1$. From this, one can see that a parabolic trajectory is best as the exit trajectory is directly in line with the body's trajectory. We can also see that as $e\rightarrow\infty$, the exit trajectory is at right angles to the body's trajectory and we get no help from the assist. Additionally, as your velocity increases, it will force $e$ to become larger unless you significantly increase the mass of each subsequent object. So, for a normal assisting body, like a star or a planet, travelling past it at relativistic speeds will result in minimal orbital deviation; there will be practically no transfer of momentum, which makes for an even smaller increase in velocity.
The fastest a spacecraft can get to using gravity-assists very much depends on the largest mass of the objects you use. However, I cannot give you an estimate of a number because due to the sheer impracticality of using gravity-assists to achieve extreme velocities, we (rocket scientists) haven't ever tried computing a theoretical limit. I can guarantee you though that without using high density objects (neutron stars, black holes, etc.), no spacecraft will reach velocities near the speed of light by gravity slingshots alone.
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I'll begin by answering the easier questions. No, Aluminium is not up to the mark. As mentioned in the comments, the spacecraft we create would probably have several stages that are jettisoned at various points in the journey.
However, given that this question seems fairly hypothetical. If we were to design a single-stage craft for interstellar travel to an Earth-like extra-solar planet out of some new material, it would have to have many of the properties you mentioned.
For starters, the spacecraft itself would have to have power generation capabilities, which would provide the heat for it when in interstellar space. However, as you mentioned, being close to 0K, the material would need a very low emissivity across the IR range, there are many white paints that can accomplish that. Additionally, if we assume that the spacecraft will be travelling at a high velocity, the material will need to be strong enough to withstand initial acceleration as well as hard enough to resist the erosion due to interstellar particles (dust, micro-meteoroids, etc.), which will be impacting it equally fast.
Furthermore, interstellar dust can have chunks up to 100g in size. At 0.2c, these can impact the spacecraft with the force of several (40 more or less) atomic bombs. So, the material needs to be extremely puncture-proof, ablative, regenerative, or Adamantium.
Withstanding the solar furnace is not an issue. If the spacecraft is moving relatively fast, it will reach a far enough distance too quick to need to worry about dissipating the Sun's heat. However, when it reaches the new star system, it will have to be able to dissipate not only the heat of that star, but the heat of entry into the planet's atmosphere. The former requires radiative cooling; a high emissivity, which is easily accomplished by dropping an outer shell and revealing a dark surface pointed away from the star. The latter can be accomplish if the material has a high thermal conduction. The heat (generated on the front side) can be transfer to heat sinks in the back and convected away.
As for the radiation along the way, an atomically dense metal should be capable of shielding a decent portion of it. Of course, it is impossible to shield all radiation due to the creation of Bremsstrahlung radiation, so the electronics would all have to be radiation hardened. For more sensitive equipment, a Faraday cage as well as radiation-blocking materials (ice, lead, gold, deuterium) could keep them more or less protected.
Having described all of these properties, I'm beginning to think that Adamantium would be perfect. Indestructible, good thermal conductor, heavy metal...