Planets appear to move horizontally when they are in that part of the ecliptic that lies horizontally across the southern horizon at midnight (seen from the northern hemisphere). That occurs when a planet is at a right-ascension of 18 hours (in Sagittarius), and a couple of hours either side. A snapshot of the planet taken every night at midnight shows it moving horizontally (approximately) from right to left. All planets do this.
As far as I know, there is no special name for this, in astronomy.
Planets move away by gravity not holding it together and also gravity holds planets together
The planets move on little circles that move on bigger circles.
Yes. All planets move in orbit around their host star.
planets appear to move
Apparent motion of the planets.
Planets naturally move in our sky over time and therefore are over different states.
If the planets did not move in their fixed orbits they may dash each other.
they move because of space eruptions
The path where a planet moves around the sun is called an orbit.
Because of the gravity
yes they do.