Why does Hanekawa know about Black Hanekawa?
Some details in the ONA for Bakemonogatari suggest a potential answer. In episode 13, Hanekawa states that she has no memory of Golden Week beyond the burial of the cat (or slightly earlier). So she can notice such gaps in her memory. Hanekawa comes to Araragi for help after growing cat ears (which came after her headaches). It's possible she also remembers that.

Hanekawa probably at least knows that something is up. Moreover, Araragi states that he has no idea how much of the events of "Tsubasa Cat" Hanekawa remembers, but that he won't ask for now because he wants to let her come to terms with her feelings.

In "Tsubasa Song", Araragi says (own emphasis, quotes from a fan translation):


  . . . it was twice that a cat had rampaged—on Golden Week and just a couple of days ago . . .


Moreover, we have this:


  And that’s why I ended up seeing nightmares not once but twice—said Hanekawa, as if telling herself that rather than me.


And later, Hanekawa tells Araragi:


  I don’t want to shut myself out anymore, and as it is, if I don’t demonstrate more of my individuality, then one day I may fall prey to the cat again.


Araragi probably hasn't asked Hanekawa about the events of a few days ago, but Hanekawa knows about both possessions. So it looks like Hanekawa got suspicious and asked someone what happened during these two periods.

It's possible that she remembers bits of the second possession or even finally remembered the first possession. But in "Tsubasa Tiger", Hanekawa comes to the conclusion that she must have been possessed at night via indirect evidence and not memory. So memory loss can be linked to possessions. Thus, it seems unlikely she remembers everything at this point prior to the end of "Tsubasa Tiger. (It's possible that she does remember everything once she accepts her repressed emotions, but that comes after the period discussed in this question.)