My understanding is that it's easier to send something out of the solar system than into the sun due to the need to overcome the speed at which the earth moves around the sun.
My understanding of planetary flybys is that the spacecraft is pulled forward when headed towards the planet and is pulled backwards upon leaving. Net gain: 0. However let's imagine the solar system as a flat disc and we're observing from the top (read north pole of earth is facing us). The spacecraft approaches Jupiter and passes it on its right side. The difference in speed in the direction it was going is still 0, however Jupiter has pulled it "left" on both approach and retreat which means the spacecraft is headed "left" in addition to its original speed. The new vector is faster than the old (though in a different direction).
First question: Can't following a planet in its orbital path produce a speed boost? The planet is moving away from you so it seems like you would 'fall' farther than if the planet was stationary.
Second question: If you fly by the planet on the 'left' side (again viewed from the top) the planet is moving towards you. You have angular momentum with respect towards the sun from when your rocket left earth. But if you fly by Mars on the left side, this momentum is lessened. Continue on and go by any other planets you can on the same side. When you hit the last planet, orbit around and head back in the direction you came again passing planets on the same side. If you pass enough of them wouldn't you end up removing all momentum with respect to the sun and eventually be able to hit it?