All of the statements are possible in this scenario:
1. The asteroid slows down as it passes around Jupiter: This is possible because Jupiter has a massive gravitational pull, which can cause the asteroid to lose some of its kinetic energy, resulting in a decrease in speed.
2. The asteroid continues out of the solar system: This is also possible if the slingshot effect (also known as gravity assist) is strong enough to propel the asteroid to escape the gravitational pull of the Sun and other planets, sending it into interstellar space.
3. The asteroid adopts an unusual orbit around the sun, such as an egg shape: This is possible if the gravitational interaction with Jupiter significantly alters the asteroid's original trajectory, causing it to enter a new, non-circular orbit around the Sun.
4. The asteroid falls into an irregular orbit around Jupiter: This is also possible if the gravitational pull of Jupiter is strong enough to capture the asteroid into a stable or unstable irregular orbit around the planet.