Question: Would it be theoretically possible to deduce the masses and distances of the other planets in our Solar System solely by observing the movement of our Sun? (Looking at the movement of the parent star is one way to detect exoplanets, isn't it?)
It is a bit harder because we are ‘in’ the Solar System, but technically yes it is sort of easy to do. This is how some of the moons of Jupiter were found before we sent spacecraft there. We can look at the orbit of the parent star and see if we can see wiggles in the periodicity of the orbit, or see darkenings on the surface that would indicate an eclipsing planet, for example.
That would be a very interesting project! It would be very complex to model with a 9-body system to the first approximation, but I’m pretty sure we would be able to see the effects on the sun from the larger planets. Maybe the proximity would allow us to measure the smaller planets too. You could probably even see things like orbital inclination of planets by looking at the motion of the sun. You could even build a time lapse of doppler shift from the sun and build up a big 3d picture of the solar system over a long time. You’re right – you can look at the wobble of a star to detect an exoplanet.