Emma Frost was recently revealed as the prime reason behind the mutant/Inhuman war. In her despair and grief over what happened to Scott, she carried on the mental illusion of what she believed he (and thus they) would do in order to protect his people from the Terrigen mist. If it meant destroying the mist and sentencing the Inhumans to their own form of extinction, oh well. When the depth of her manipulation (and subsequent mental break over what happened to Scott) was revealed, she escaped and has since become hunted by the various X-teams.

Havok was a victim of the Scarlet Witch's moral inversion spell back during the AXIS event. Wanda,along with Strange, had cast this spell in an attempt to stop Red Skull (who still possessed Xavier's brain) who had then become Red Onslaught. Before this, Alex was a part of the Avengers' Unity Squad and under Remender, had actually married Wasp and the couple had a child named Katie. But after suffering the spell, Alex's morals changed, fearing what the Avengers would do with the now subdued Red Skull. He quit the Unity Squad over these differences and reconciled with his brother Scott, with whom Alex now shared his beliefs about mutant endangerment and possible extinction. (And also apparently abandoned Janet as well. The fate of Katie is another tale).

Later, Havok never believed that Black Bolt killed Scott. When confronting Emma about it, she eventually confessed to him her grand deceptions concerning Scott and fostering war with the Inhumans. When her deception was revealed, Alex helped Emma to escape out of respect of his brother's memory (apparently). So to date, Alex has been working with Emma.

For some odd reason that I can't quite recall, Havok and Sabretooth are the only inverted characters still active since AXIS.

As for Miss Sinister, she is the current reincarnation of the Essex personality.  

One way to measure the success of a new creative team: What new elements do they bring to the mythos? Do they just churn existing characters and concepts? Or do they add something that future writers can build upon?

The description above sounds like a lot of churn.

Yep! Contrast with what X-Men were doing as recently as 2001-2006 (which is over a decade ago, I know) with Grant Morrison and Joss Whedon, who introduced:

Cassandra Nova
Emma's diamond mutation
John Sublime and The U-Men
Quentin Quire
Kick
Beak
The Stepford Cuckoos
Negasonic Teenage Warhead
Fantomex and E.V.A.
Weapon XI and higher
Dust
Xorn (which... I know...)
Kavita Rao and the mutant cure
Ord
The Breakworld
S.W.O.R.D.
Agent Brand
Danger
Armor
Blindfold

...Cyclops as a bad ass!

First of all, I will immediately plead guilty to not having read all the X-Men from any of the last three decades. I have read the Hope Summers origin stuff (including Bishop chasing Cable around through time) and the series where a zillion Nimrod sentinels attack the X-Men island. Not a lot else in the last decade.

When I said "churn," I was thinking of soap operas, where what passes for plot development is everyone trading romantic partners, then starting the whole sequence of temptation and infidelity again. The Hellfire Club could be churn if it was just "another group of evil Mutants attack the X-Men"  But it was really just the circumstances of the real story, which was Jean Grey's loss of innocence, addiction to power and passion, and eventual self-destruction, which is very much character driven rather than churn.

"Xavier's School has an extremist student who develops a worrisome cult of personality among his peers" was a non-churn Morrison story. "An alien race is convinced that Colossus is going to blow up their planet and decides to blow us up first" was a non-churn Whedon story. The fact that these stories could be described as "Mutants vs Mutants" and "Mutants vs aliens" does not really tell the tale of either the plot details, the themes unfolding, the new plot elements, or the characterization.

Marvel has always gotten mileage out of having its dysfunctional heroes fight each other, then reconcile and fight a common foe. Early FF/ Avengers/ X-Men crossovers saw this. Contest of Champions, Secret Wars, Civil War, etc. Easier than coming up with a new character, right? But it gets old if all that we see is new combinations of the same heroes taking sides against each other. Having not read the stories: "The X-Men split into two rival factions who fight" sounds like churn on the surface. "Mutants and Inhumans fight" sounds like churn. Maybe there's a great character driven X-Men 129-137 lurking in there too; I don't know. Are there new characters and concepts?

Part of the problem, as I see it, with the X-franchise at the moment is the lack of focus on new characters. Morrison and Whedon did introduce new mutants who were the focus later on in such titles like Kyle & Yost's New X-Men/Academy X-Men. But since their departure, these new characters quickly fell off the radar. The biggest request/complaint that I've heard from other X-fans is for Marvel to give proper attention and new focus to these under-used mutants.

Even when Marvel juggles the line, their natural inclination is bring back the original New Mutants (Sam, Bobby, Dani, Illyana, and such). Characters who by all rights, should be full-fledged X-Men already. So there's room for focusing on these new kids as new students. Then, there's the writers to consider too. I don't know how accurate these site interviews are with them when they take over a new title, but they often say that editorial gives them an option of which characters they want to use and more often than not, it's older characters they tend towards because they are the ones of whom they'd follow in their youth as well.

Even Bendis fell into this motif when he took the reigns of the franchise. Instead of using the previous new kids, he created his own new characters instead. It's a self-destructive cycle that revolves around the popularity and clout of the classic characters, even the 'kids' of the New Mutants. That's why I love Claremont's underlying belief that these characters should be allowed to grow, age, retire, or even die so as to make room for new characters and new drama.