Is it probable that planets will stop orbiting in ellipses? I saw this video, but I had the impression that orbiting in an ellipse or a circle is quite a coincidence - it depends on how the planet got into the Solar System. Because I think that either all active forces balance in a circle, or simply all the planets with too much eccentricity burn in the Sun, or to break away from the Sun's gravitational field. Is it true?
 A: A planet orbiting in a true circle would be an extreme coincidence, but there are infinitely more ellipses than circles.  
However, an ellipse is not a coincidence.  Any object that doesn't have enough kinetic energy to esacpe the gravity of a star will have an elliptical orbit (in the approximation of Newtonian gravity, no additional objects, spherical shapes).  
If there is more kinetic energy than needed to escape, the path would be hyperbolic, not elliptical.  
The possiblity of burning up in the Sun does happen to comets sometimes.  
A: In a planetary system you would expect the orbits of the large bodies to be almost but not quite circular. This is because there are two opposing effects: tidal forces tend to make orbits circular but perturbations between planets tends to make orbits elliptical.
In a two body system, just the star and one planet, tidal deformation of the planet and the star will dissipate energy and the effect of this is to make the planet move outwards. Exactly this effect is currently causing the Moon to recede from the Earth at around 4 cm/year. The tidal forces vary strongly with distance, so in an elliptical orbit the tidal dissipation is much greater when the planet is nearest the star than when it's farthest away. The end result is the the orbit gradually becomes circular.
With only two bodies the orbit will remain circular, however when other planets are present the planets all exert gravitational forces on each other and they perturb each others orbits. The end result is that the orbits become slightly elliptical, and also also that the eccentricity of the orbits is continually changing. For example the eccentricity of Earth's orbit varies between 0.000055 and  0.0679 over a period of about half a million years.
Small bodies like asteroids and comets can have wildly elliptical orbits, but over the long term such orbits aren't stable because close approaches to large bodies like planets are likely to eject the asteroids from the Solar System. In general stable elliptical orbits are only possible when they are stabilised by a resonance, for example as Pluto's orbit is stabilised by a 3:2 resonance with Neptune.
