I have found that sometimes (not all the time) when i get pulled into the First star's gravitational pull i get pulled into the very center of the star.
Actually this happens when an npc interdicts me while I am so close to the star.
Yes it does.
New Engineer coming (ConfirmedNotConfirmed): Sunny, McSunFace - in order to get him to like you, you have to get pulled into the primary star's centre during an interdiction.
He upgrades your heat sink.