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It is known that all the mass in the Solar system moves around the Barycenter.

For the two focal points in Kepler's law; is the first focal point $F_1$ a Heliocenter? Or is it actually a Barycenter?

If the $F_1$ is a Heliocenter, then, can the same exact ellipse and eccentricity of orbit be maintained by adjusting the second focal point $F_2$ and taking Barycenter as $F_1$ instead of Heliocenter?

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Kepler's laws and the associated orbit only hold for a two-body problem, so the 'barycenter' in the question can only be understood as the center of mass of the sun and a particular planet (ignoring the other ones), not as the solar system barycenter. And in this sense the convention is to take the sun at the focus, which means the semi-major axis is the (maximum) distance between the sun and the planet.

If you define the orbits with regard to the center of mass of the two bodies, the semi-major axis is reduced by the mass factor $m/(M+m)$ for the planet and $M/(M+m)$ for the sun whilst the eccentricity as well as the orbital period stays the same. However, with the center of mass only a fictitious point that does not correspond to a physical object, this hardly makes sense in practice.

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Planetary orbits are normally described as heliocentric, but it is possible to describe them from a barycentric point of view. JPL Horizons (https://ssd.jpl.nasa.gov/horizons.cgi) provides for both possibilities.

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It depends!

If we plotted the orbits of the innermost planets in our solar system, they would be closer to ellipses with the Sun at one focus.

If we plotted the orbit of Jupiter, the biggest "gravitational bully" in the solar system (it messes with everything!) it would be closer to an ellipse with the Sun-Jupiter barycenter at one focus.

It gets a little complicated out there because the next three planets (Saturn, Uranus and Neptune) also push the Sun around a lot. We might think that lightweight Neptune wouldn't do much, but it's larger distance makes up somewhat for its smaller mass, since the center of mass is weighted by the distance*mass product.

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    $\begingroup$ +1 If one models the solar system composed of the sun and eight planets, is there a way to relate the focus of the orbit of one of those planets to the barycenter of the system? Would it merely be a change of reference frame? $\endgroup$ Jul 6 at 14:01
  • $\begingroup$ @DaddyKropotkin I think so. To first order Jupiter's effect can be thought of as far enough away and changing slowly enough that it accelerates the Sun and the inner planets in a sufficiently similar way that the inner planets move about the Sun in nice elliptical orbits even though as a group they all dance with Jupiter without "knowing" it. $\endgroup$
    – uhoh
    Jul 7 at 0:51

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