There are some leading Jupiter and some following. They are collectively known as the Trojans. Some asteroids are rogue in nature and orbit the Sun at many different distances and can have circular or elongated orbits. Some of these asteroids cross the Earths orbit, but it is most unlikely that we will see one hit the Earth in our lifetime. In fact it is unlikely the Earth will be hit be a large asteroid for millions of years.
Comets are different in nature to asteroids.
When the comet is close enough to the Sun the ice melts and the gas and dust is forced into trails. The gas is directly behind the comet with relation to the Sun, and the dust trail is to the side. A diagram is shown below of this relationship. The dust follows the orbit from where it breaks of and so the trail appears to be a large curve. Most Comets are thought to reside in the Oort Cloud which is far beyond the orbit of Pluto.