If I’m not mistaken, such a trajectory would pass through or very close to the Sun.
Which would nicely match the visuals: Kirk departs Earth with the planet's sunny side receding in the viewscreen.
No exactly. In the original edit the planet is partly in shadow (http://movies.trekcore.com/gallery/a.../tmphd0760.jpg
), and in the DE it's mostly in shadow (http://movies.trekcore.com/gallery/a...icture0420.jpg
). In either case, the direction is wrong for heading too close to the sun.
Then again, as neither planet's orbit is perfectly circular, and their aphelions aren't aligned, maximum distance between them isn't centered right on the sun. You'd fly near it, but not necessarily super close or through it.