So that all the elliptic orbits can be seen as slices from the cone while the sun is a shared focus.
Almost, but each orbit corresponds to its own imaginary cone. Planets have different inclinations and directions of the plane.
Yes and no.
What you seem to really asking is about the idea that the four types of orbits a body can have (circular, elliptical, parabolic, and hyperbolic) are also conic sections. In an ideal case where you use Newtonian physics and have just a single planet and single star, yes you'll get those conic sections. However, the planets in our solar system do not follow orbits which are perfect conic sections for a few reasons.
- Our solar system has several planets in it, all of which are affecting the orbit of the others. While the Sun is the predominant factor in any planet's orbital parameters, the other planets have non-zero, measurable effects. For the Earth, the perturbations from other planets (primarily Jupiter and Saturn) is described by the Milankovitch cycles which are measured variabilities of the Earth's orbital parameters. What this means is that, at a fine scale, the Earth's orbit (as well as the other planets) is not a perfect, closed ellipse and does vary slightly from a conic section.
- Newtonian physics is great for most scenarios, but in some cases, General Relativity starts applying. A specific example is Mercury. Mercury's orbit is greatly perturbed by the other planets, as described in point (1), however there are additional relativistic effects that cause Mercury to have an unclosed orbit. This effect is usually called the perihelion precession of Mercury and results in an orbit that differs quite significantly from a standard conic section.