What is Caroline's fate at the end of Portal 2?
GlaDOS is made up of a few different personalities (the personality spheres), and Caroline was probably one of the ones that was integrated closest to GlaDOS. It's fairly likely that Caroline's more closely integrated and influenced aspects of GlaDOS's development, but GlaDOS clearly has her own personality because otherwise she probably wouldn't have flooded the enrichment centre with a deadly neurotoxin.
There's a much simpler explanation for the turrets ignoring Chell; GlaDOS no longer wants to kill Chell, so the turrets no longer recognise her as a target. Because they're made by Aperture, though, they're quirky enough to recognise Chell as an audience instead. Hence the singing - you can run across them practising earlier in the game.
I see the 'GLaDOS is Caroline' thing as GLaDOS is a personality core that contains what is left of Caroline. She went crazy because 1) the power of the GLaDOS body makes you a little crazy and 2) It's hard to go through that much and still be who you were. she adapted to that body but it took some work and time; I'm sure her personality changed over that time. After walking around with Chell, GLaDOS remembered who she was before she was GLaDOS and as a result is more 'human' but still knows how to deal with her body so she is here to stay.
A further note on Caroline changing when she became GLaDOS from http://half-life.wikia.com "The system rewards the administrator of constant repeated tests with a euphoric reaction upon test completion, much like a drug addiction. The euphoria diminishes over time, causing a state of withdrawal which can induce insanity. GLaDOS claims that she was sufficiently motivated by a earnest love of science to overcome the withdrawal, but it may still have had effects on her personality. She also describes the maddening influence of the constant voices provided by the personality cores used to suppress her homicidal compulsions." I see this as further evidence that when Caroline became GLaDOS she had major personality changes
As for the 'deleted Caroline' thing I think this is just part of GLaDOS's sense of humor. She frequently lies and mocks Chell for her own amusement. Also this could be seen as a similar thing to someone saying "Just go! I don't want you here anyway!" when they say good bye to a loved one.
To add to Merus' answer - GlaDOS has already shown she can function just fine with personality cores removed; see the boss fight in Portal 1, and Wheatley-the-tumor in Portal 2. If the "Caroline" personality was the original core (and why not? Aperture Science seems real big on Modularity), there would be nothing to prevent GlaDOS from removing it, just as she "convinced" Chell to remove the Morality Core in Portal 1.
The other option, of course, is that GlaDOS is lying about having deleted Caroline. She's not exactly got a great record as the most truthful of sardonic, all-powerful AIs.
Either way though, it's all just speculation. The only ones who would know for sure are at Valve.