It is impossible to find an exact solution for the motion of all the objects in the solar system. In fact, it's impossible to find an exact solution for just three objects in Newtonian gravity. Therefore, all we can do is to express the motion in approximate terms.
However, we can make some very good approximations, because the Sun is so much more massive than the other objects in the solar system, and therefore as a first approximation we can consider each planet as independently orbiting the sun, ignoring the interactions among the planets.
To a first approximation, the Earth orbits the Sun, which is still.
A better approximation would be to acknowledge that the Sun and the Earth both orbit their common center of mass (barycenter) on ellipses. Of course, the Sun barely moves, while the Earth travels on large ellipse.
We can increase the level of sophistication by including the effects of other planets in the solar system. The other planets will shift the barycenter of the solar system. They will also exert additional gravitational forces on the Earth. Accounting for these small effects was one of the first applications of perturbation theory in physics.
Today, we rely on precision measurements and computer models to get the best available description of orbits in the solar system.
Anyway that was a tangent. To return to two specific points you raised:
In this case, is the barycentre a fixed, stationary point or does its position depend on the position depend on the planets' positions (disregarding outside factors: assuming it is a closed system).
In the solar system's reference frame, the barycenter is a fixed point (assuming no external forces act on the solar system).
Moreover, if the Earth's orbit is an ellipse, what causes the Earth to accelerate when it passes the aphelion and decelerate when it passes the perihelion?
Kepler's third law dictates the speed of the motion of the Earth on its (approximately) elliptical orbit. In modern terms, we would phrase this as conservation of angular momentum. Briefly, the gravitational force scales as the distance between the Earth and Sun to the minus 2 power, and so the Earth feels a stronger force (and a larger acceleration) when it is close to the Sun and a smaller force (smaller acceleration) when it is far from the Sun. At the periheilion, by definition the Earth is at its point of closest approach to the sun, so its force (and acceleration) is at a maximum. Therefore as the Earth moves away from the perihelion, the force on the Earth (and therefore its acceleration) decreases. The same argument (in the opposite sense) applies to the aphelion.