Hypervelocity stars are created through close gravitational encounters with other bodies, which are likely to strip away planets, especially those particularly far from the star. You have declared that the planet is "stable", and I'm happy to work with this, but that means you can essentially choose what happened to the system's planets and moons, and that's all that will matter here.
Seasons have nothing to do with the interstellar environment or the star's motion relative to the galaxy — they're a function of the planet's axial tilt and its orbit around its star. Since star systems in our galaxy are already only negligibly affected by forces from outside them, and this system is outside the galaxy and thus even further away from any outside influences, whatever's influencing seasons will be internal to the system. Just as in any other case, a moon may help stabilize the planet's axial tilt and keep its seasonal strength stable, while other planets in its system might perturb its orbit over millions or billions of years. Exactly what is in the system to influence these is entirely your call, since you've said that the result is stable and habitable by fiat, and has nothing to do with the star's hypervelocity nature.
The star's velocity also doesn't matter for anything related to energy balance or zodiacal light. Intergalactic space is essentially empty; the numbers typically thrown around are on the order of 1 proton per cubic meter. The total energy of these particles on a planet sweeping through at typical-to-high hypervelocity speeds of 1,000 km/s is roughly 1 nW/m^2, which is much fainter than Earth's zodiacal light from solar system particles and would be totally drowned out in a similar system even before accounting for the star's heliosphere or the planet's magnetosphere protecting it.
In intergalactic space, there will probably be no visible stars. The Milky Way and Andromeda, and possibly other small Local Group galaxies, may appear in the sky as fuzzy blobs, allowing possibly for some amount of celestial navigation. Because the "night sky rotating" is a function of Earth's rotation and its orbit around the Sun, and a star moving quickly through space does not directly impact either of these things, one would expect the exact same cycle — this planet's star rises and sets as it spins, and the night sky is always visible on the other side, while over the course of its orbit the visible half of the night sky sweeps out around the ecliptic in a circle.
Andromeda's half of the sky will be slightly blueshifted, by an increasing factor centered towards Andromeda — if it's approaching at 1,000 km/s (1/300 c), then it'll be by a factor of about 1/300, or about 2nm in typical visible wavelengths, enough for spectrometers to easily notice but not enough for a human to tell. The Milky Way's half of the sky will be similarly imperceptibly redshifted.