As mentioned in this answer the world in starbound loops around.
Can I keep digging down and end up on the other side of the world? What happens when I keep digging down?
If you dig down, you will reach bottom. Your screen will stop scrolling.
No, you can't. You can walk around a planet though, so when you walk to the west or east for long enough, you'll once again arrive at your starting point. As far as I know you'll hit an unpenetrable border, so the planets are essentially rings; at least when considering the accessible parts.
No, you can not go straight through, when you are near the bottom, there is a giant magma pool. This is the core. You can use blocks to keep going further, but this is not recommended, as when the screen stops scrolling, if you touch the bottom of the screen, it is insta-death.