You're correct that space is not completely empty and contains various particles, including atoms and molecules, albeit in extremely low densities. However, at near light-speed velocities, the main factor affecting a spaceship's heating would not be friction with interstellar particles but rather a phenomenon known as "relativistic effects."
As an object approaches the speed of light, relativistic effects become more significant. One of these effects is time dilation, which means that time appears to pass more slowly for objects moving at high velocities relative to an observer at rest. Another effect is length contraction, where objects in motion appear shorter in the direction of their motion when observed by a stationary observer.
Regarding the heating of a spaceship, the primary challenge at near light-speed would be dissipating the enormous amounts of energy generated due to the ship's motion. As the ship approaches the speed of light, its kinetic energy increases significantly, and the energy must be managed to prevent overheating and structural damage.
In reality, achieving near light-speed is an enormous engineering challenge due to the tremendous amounts of energy required and the relativistic effects that come into play. The concept of achieving such speeds while avoiding obstacles and managing energy remains largely theoretical.
It's worth noting that Star Trek's warp drive, as mentioned earlier, circumvents the issue of friction and heating by distorting space-time rather than propelling the ship at near light-speed.