Voyager 1 spots new region at the edge of the Solar System

You could look up the planned mission duration, but all this would tell you would be the duration that they were absolutely designed to work for -- kind of like how the Mars rover missions were 90 sols, but in actuality they've gone over 2,000 (albeit with solar panel dust issues, busted wheel motors, etc).

Yes, this is basically correct. What a gravity boost from a planet does is transfer a very small amount of the planet's orbital kinetic energy to the spacecraft; it puts the planet into a slightly smaller orbit, and boosts the spacecraft to a higher speed on its way out. Since the Sun is what the planets are orbiting, this trick obviously won't work the same way with the Sun.

(Technically, the Sun itself, or more precisely the center of mass of the Solar System, is orbiting the center of the galaxy; so I suppose it would be theoretically possible to fly a spacecraft by the Sun in such a way as to transfer a very small amount of its orbital kinetic energy with respect to the center of the galaxy to the spacecraft, so it would fly outward faster than it went in, even after taking into account the slowdown climbing out of the Sun's gravity well. But I doubt we're going to be in any practical position to try this any time soon.)

A gravity assist works by robbing orbital momentum from the planet, transferring it to the spacecraft. (I once saw a calculation that Voyager's slingshot slowed Jupiter's motion by one foot per trillion years.) Jupiter has orbital momentum around the Sun, but the Sun has no orbital momentum around itself.

Flying past the Sun could give you velocity relative to the galactic center, making use of the Sun's orbit around that. But that velocity doesn't help you leave the Sun's neighborhood or travel within the Solar System, because the gained velocity is in the same direction as the Sun and Solar System are already moving.

In the reference frame of the planet you will travel in a hyperbola as you go by. The incoming branch is from the front of the planet, the outgoing branch is also in the front of the planet. So you go from moving backwards to moving forwards.

However the planet is moving in the reference frame of the Sun. The initial "moving backwards" is actually more like "sitting there". The ending "moving forwards" is actually something like, "moving up to 2x as fast as the planet".

So a gravity boost only looks like a gravity boost in a reference frame that is moving relative to the object. Furthermore the gravity boost also does nothing to help you to escape from the object you're getting the gravity boost from.

Therefore we cannot get a gravity boost from the Sun to leave the Solar System. And a gravity boost from the Sun does not look like a gravity boost to us, in orbit around the Sun.