OMG, you made my day on SO many levels!!

I have those same ships! I've made up an AU for them, but I've done a more recent bio of it, which is probably cleaner typing-wise than my original post of it on this topic.

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First off, take a look at Beast Boy's & Terra's biographies, which define how they got their powers, & everything else.
Beast Boy's:
Page 1,
Page 2,
Page 3,
Page 4,
Page 5Terra's:
Page 1,
Page 2,
Page 3,
Page 4,
Page 5,
Page 6Also, Beast Boy's wikipedia webpage bio, I use some of it too:
http://en.wikipedia....acter_biography-----------
How they metWhen Beast Boy's family were still in South Africa (none of those bios state where, but I've seen it mentioned on a fansite's bio. I forget which one though.), just a couple months BEFORE his parents' deaths in the flood, BB spots Terra while the 3 of them were exploring the area together. Similar events to the first animated Terra episode allow her inability to control her powers to be revealed, although his parents are there to witness it, & thus never a secret. Beast Boy intially convinces his parents to allow her to stay on as a house guest (rather than an adopted sister as he's already devolped a crush on her, & secretly vice versa), but she gets scared as in the series over one thing or another & attempts to run away, but due to the age ranges his parents convince her to stay permanetely, promising to only help her with her powers & virtually everything else, treating her as a daughter despite not adopting her.
His parents stay aliveTerra BARELY manages to save the 4 of them from the flood (a similar manuver to how she begins the training session with the Titans in the first cartoon appearence), & shortly after the family decide to move back to the states (not far from Doom Patrol HQ, yet also in Jump City near enough for the two kids to go to the school Terra's lookalike attended in Things Change).
His kidnapping, & Doom PatrolBeast Boy's kidnapping & such still happen, also a primary reason to return to the States. Terra assists in his rescue, (I've read on a BB site/fanfic that its the Batman who almost rescues him. Terra learns of this while looking for her friend/crush & joins in on the rescue) causing them to become even closer upon the reuniting scene. Possibly a year later if that long, Beast Boy learns of the Doom Patrol and convinces Terra that is how they'll learn more about, and how to control their powers. Similar joining occurs, & Mento treats Terra as basically the opposite of how he does Beast Boy, being able to slowly but surely gain faster control of her powers & similar reasons.
After a few years in the DP, Terra & Beast Boy become a couple around age 10-11, most likely during that Christmas after a heroic rescue on his part, & a great understanding of both their powers and the control issues. (think another brain switching)
The Teen TitansWhen Beast Boy is rejected from the Doom Patrol in the beginning of Homecoming: Part 1 (the flashback before the theme song), Terra joins his leave from them stating she wouldn't consider herself a supportive friend and girlfriend if she stayed on, especially considering Mento's favoritism of her. About a year or two afterwards, the events of the "Go!" episode (what should've been the pilot, how the 5 Titans met & became a team) occur, BBxT still a pair as ever & likely on their first true date, being old enough to go out after dark without his parents. The Titan Rising episode's events happen shortly after "Switched", giving Terra & Raven the ability to consider each other buddies, sharing books & secrets & whatnot. In fact, Terra is the only one to discover Raven's secret crush on Aqualad after his first episode, first detecting the hints in said ep, and her literally grabbing her & taking Terra inside Raven's room to tell her. This is needed for the Aftershock episodes.
Episodes Betrayal, Aftershock Parts 1 & 2Speaking of which, Terra's betrayal and joining with Slade NEVER happen. Instead, Slade becomes more & more interested in merely the girl's POWERS, making the Betrayal episode happen with a
clone of Terra, which the Titans save for Beast Boy (who is on the date with the real girl as seen in the original episode) truly believe its her. They're sure of this due to the clone Terra showing up while the real deal is off somewhere with Beast Boy &/or his parents, & dropping pretty decent hints that she's done such betrayal including destroying Raven's trust with the real Terra overtime. No one learns of her secret crush on Aqualad, but the clone makes Raven believe she has.
Upon Beast Boy's & Terra's return to the tower that night finding their left-behind communicators at home, the two are needless-to-say shocked beyond belief, and literally chuck the communicators at Robin & Raven respectively, stating their leave as Titans. This isn't resolved until Aftershock Part 1, in which the two lovebirds discover the Terra clone at the same time as the Titans, Beast Boy's parents on the scene as well to witness it all unfold. everything happens as is afterwards, EXCEPT the clone Terra betraying Slade. The turned-to-stone event happens when Terra becomes SO enraged at Slade for everything, destroying both him & her clone in the process of triggering the volcano.
Beast Boy's depression; Terra's returnNaturally Beast Boy is blown back a 1,00% over his only girlfriend's (& first FRIEND period.) death, thus either being a completely depressed/ANGERED mope in some episodes, (Beast Within for example) & not present at ALL in others.
This stays intact until The Beast Within, in which Raven of all people is so frustrated with the boy's attitude that after said episode occurs, she makes it her life's MISSION to bring Terra back from stone. She finds the same spell used to restore the universe to its proper state (including returning people from stone) in "The End: Part 3" (the ending of the episode) and uses it on Terra succesfully, alerting Beast Boy's mother so that the woman is present to help her & Starfire. His mom calls Beast Boy the second their sure the spell worked, & needless to say BB is there the second his body can get him there, and Terra is greeted by him holding her in his arms, both his parents & all of the Titans West present with him.
It DOES take Terra awhile to get her legs back in proper working order, so she is hindered for a good few episodes after, yet still manages to fight with the other Titans, Beast Boy just a LITTLE too overprotective of her, gee yathink?? =P!
Slade's returnWhen Slade returns, oh GOD are the two ticked, in fact its impossible to tell who's more out to kill Slade with everything they've got because everything's so intense. Trust me you DON'T want to be in that room! So after everything, Slade also becomes their personal lifelong enemies, just as he is with Robin.
Reuniting with Doom PatrolWhen Homecoming episodes Part 1 and 2 occur, Terra is still hindered a bit with her legs, so the first part's ending where Beast Boy has to choose between saving everyone & stopping the Brotherhood of Evil, he doesn't even TRY to take time. He hears Terra's groaning over her legs when she falls almost (Rae keeps her up inbetween attempts to break out), & he literally starts moving, their out maybe 3-30 seconds later. This of course causes Mento & the Doom Patrol to learn the short version of what happened to Terra, & why Beast Boy's that danged protective of her. Terra's powers DO avoid the Doom Patrol's seemingly destruction in Part 2, & she discovers metalbending back when their first attacked after they find RobotMan, enabling her to help & nearly be the true hero of the day. In order for it to still be BB's episode however, he's still the one to save everyone at the tail end, as well as getting all the bonus points with Mento in the two episodes.
Final Episodes, Titans in TokyoBeast Boy & Terra find each other quickly during Titans Together, & spend the remainder of the episode shown as a couple so close NOTHING can break them apart, anytime the camera focuses on them. Needless to say "Things Change" NEVER happens,
EVAR. Titans in Tokyo has Beast Boy swarmed by the fangirls, causing him & Terra to have a small argument after the scene, but it was HER that talked him into going up onstage, rather than chasing the schoolgirl because Terra is the only girl Beast Boy sees in this AU. This allows Terra to see quickly enough that it was her fault in the first place, the scene has Beast Boy only trying to be nice to the other girls, getting Terra's attention so that she knows he desprately wants her to help him out of it from the first second. They ARE resolved by the time the Titans rejoin Robin & Starfire, & the movie resumes.
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That's it! As for Rob/Star, I've created two future children for them in an AU to the canon of the DC-verse in which they stay together, both stay alive, & all that good stuffz.

John Grayson the 2nd after his grandfather, & Ruby Grayson. I knew there was an DC-verse AU daughter of theirs called Mari' Grayson, hero name being NightStar, but I've only discovered her story AFTER creating, & placing Ruby with Damian Wayne (who HAPPENS to be paired with NightStar in that DC-verse's version of him!) in the current Batman series. Needless to say my spazz-attacks that night!