Just to address this one point:
If it falls from beyond the solar system, shouldn't it have the solar system escape velocity when it passes Earth?
I can see what you're thinking of, but the answer is "no".
If the object were to travel in a straight line towards the Sun, attracted by the Sun's gravity, yet somehow miss it and travel straight on beyond it, then at a first approximation it would have the same speed at each point behind the Sun as it had at the same distance when it was approaching it, and at the limit it would tend towards its escape speed: it would perform simple harmonic motion, like a spring. Even a spring, of course, in reality performs damped harmonic motion, because of friction, and so would our planet, because space is not empty (even ignoring its collision with the Sun itself!).
If the object travels into the Sun's gravitational field so as - initially - to miss the Sun, it would have a parabolic or - possibly - hyperbolic orbit around the Sun, and whether it escaped again from the Sun's gravitational field would depend on the velocity with which it entered that field in the first place, and on its interaction with the Sun and the remainder of the solar system - as you suggest, a slingshot effect might occur.
at the limit it would tend towards its escape speed
Just to clarify what I mean by this: the magnitude of the escape velocity at any time depends on where the object is at that time: it reaches a maximum at the centre of mass of the solar system, and decreases as the object travels away from the Sun. So as our object performs its spring-like oscillation about the Sun (with the Sun somehow helpfully stepping aside each time it passes, without disturbing the field!), its velocity tends towards that maximum as it passes the centre of mass of the solar system, and decreases to become zero at the maximum extent of its amplitude.
How extremely stupid not to have thought of that - T.H. Huxley