> Question 1 -
Well yes. If it wasn't, then the earth/jupiter/etc. would fling themselves out of the solar system because the sun couldn't hold them.
>Question 2 -
That's exactly what would happen.
>Question 3 -
Depends on what the core is made of. The Earth's core would be cold and solid except it has enough radioactives in it that the decay keeps the core molten. Heat loss isn't a huge factor when you have a slow processes like decays keeping things warm. The core could easily still be molten (but smaller! and cooler than otherwise) after a million, or billion, years in deep space.
>Question 4 -
Deep space is pretty empty. But there are other rogue objects that have been thrown out of solar systems, dense gas clouds, and other star systems to encounter. Probably the planet's crust would have cooled (and contracted) enough that major tectonics aren't still ongoing, and with no gaseous atmosphere (and rain/etc.) to erode things, you're going to have pockmarks all over the place, probably similar to the southern hemisphere of Mars, which still bears lots of scars even after a billion years.
Depends on the biology of the planet. There are alternate biologies that can survive that and are relatively resistant to cosmic rays. A liquid planetary core can still have a dynamo in it, creating a magnetic field to shield the planet of the worst of that sort of thing. I'd give fair odds that certain creatures could go dormant long enough to survive an interstellar voyage (bacteria, fungi, very simple and hardy creatures like Water Bears or whatever they're called). You're not going to get large or intelligent entities to survive, probably, unless the planet is built around a silicate kind of life, or something else that's potentially hardier than our protein-based Carbon life on Earth.