Looking at the handy cosmology map on page 161, we see what looks like a spherical earth flanked by spherical other-earths: The Shadowfell and Feywild.
I have always treated the Demi-Earths to be separate plane, and only allow planar travel to them. However, I must define parallel worlds differently than others. In my opinion, a parallel world has an alternate person for each Earthling (Like in the Doctor Who episode 'Age of Steel,' where Pete is still alive, Jackie's divorced, The Doctor doesn't exist [or is dead] and Rose is a dog). From what I can tell, the Demi-Earths are more like copies of the locations, but not (necessarily) the people (the 4e Tomb of Horrors handily illustrates this).
In summary: No, I don't think you can get to other planes by getting on a spaceship; Yes, Earth (or Toril, Oerth, or whtever else) is a planet (sounds a little stupid, actually, when I put it like that), and I don't think that either Demi-Earth is necessarily parallel.
Yes, there are stars, or else how would there be a Star Warlock Pact?