First off I think The Manga of Takeuchi Naoko has all of the books online in their original form & yes Star Saturn is her eternal form. Next her history. Now assuming you're talking about the anime version noone knows why she was reborn on earth, but we can automatically assume to be there to save Rini & the others from Pharaoh 90.
I've been pondering the question myself for years now & I can't come up with much of an explanation for why she was already there except that she woke up & dropped her glaive like in the manga in reaction to Beryl & Metalia. We don't see her anywhere on the Moon Kingdom in the anime so this is hard to believe but there's no other reasons given.
As for how she was reborn, just like the other Senshi reincarnation.
She was with the Death Busters in the S because her father, Tomoe, made a deal with the thing posessing him to work for them for Hotarus life. Mistress 9, the chick with outrageously long hair claiming to be the Sovereign of Silence, was put inside her body allowing her to live. Mistress 9 is evil, but her original soul of Sailor Saturn is there & begins to wake up towards the end of the season. She breaks free from Mistress 9 & sacrifices herself once again to protect the universe from Pharaoh 90. She is reborn as a baby & the Senshi of Death goes back into sleep or whatever you want to call it until the Stars season when she awakens again.
So did that clear it up? Her first costume is the original with the jagged sleeves & the, tasteless in my opinion, Super outfit with the long back bow that all the Outter Senshi get in Stars (the rest recieved theirs in Supers)

this is the original
heart 3nodding 
this is the super

& this is the eternal