>Assuming you drifted long enough to get away from the sun, would it look like our night sky, with hundreds, thousands of shiny dots everywhere you looked?

And could we see galaxies, nebulas and other space phenomena or are they too far apart, too big or in a spectrum we can't perceive?

Dont really need to get away from the Sun. Once you are in shadow you do not recieve reflected back glare from dust in the air like happens on Earth. You will be in pretty dark space.

You can see Andromeda, a huge galaxy a couple of hundred light years away. And the Mangellic Clouds, satellite galaxies of our own. So you can see a couple of galaxies.

I dont think you can see nebula with the naked eye. They tend to be gas clouds that only resolve with telescopes.