Nothing is permanent in this world. Galaxies dies, starts dies and the universe will not remain for eternity. The bigger starts, larger than the sun die in nova and supernova explosions when their fuel ends. It is estimated that the whole life of the sun is 10 billion years. It has spent half of its life. When it will burn all of its hydrogen, it will start fusion of helium. Ultimately, it will become a red giant and engulf planets one by one. Life from the earth will have been ended much earlier than that. So, the planet earth has maximum five billion years to survive. But, it can end much earlier than that time. Collision with a gigantic asteroid can destroy it anytime. Also, our whole solar system can be destroyed by black holes. Even the whole universe will ultimately collapse in a single point and die. This may be the restarting of the whole universe again.
So, everything has its maximum life. After a definite age everything dies. Planet earth will also follow this trait and eventually destroyed.
The earth has a limited lifespan. All of the planets in our solar system have a limited life, the end of this life may be millions of years but it is finite, it does not go on forever.
Eventually our sun will go supernova during this process it will expand massively to encompass the orbit of the earth. This will also effect the orbits of other planets in the solar system. A lucky planet to escape this wrath might just be on the right orbit to get projected out of the solar system. Failing that, the orbits of all other planets will change and a great chance the sun will consume them through the attraction of gravity.
No the earth is not eternal and will fall with time. The very thing which provided us with life, will eventually kill the planet it made habitable.
Probably not. If theoretical predictions are correct that protons have a half-life then nothing is "eternal" in existence. Even if the Earth survives the sun becoming a white dwarf and the later collision between the Milky Way galaxy and Andromeda; over enough given time enough small events will occur to break the Earth apart. We don't even know what "The Great Attractor" is that is pulling on our galaxy and other nearby galaxies so that may or may not also destroy the Earth eventually.