Yes both of these beauties are Karma. Yes she does come in different rarities. Yes it does matter how much of the event item you collect. Yes you can get the black one without spending any SC if Aigis does not hate you during the event. Yes you do have to wait until after the event to get Karma.
Requirements
250: Vampire Princess Karma joins you.
270: Karma starting level is 5
300: Karma gains -1 cost (Total: -1)
320: Karma gains 1 skill level (Total: 2/5)
340: Karma starting level is 10
370: Karma gains -1 cost (Total: -2)
400: Karma starting level is 15
450: Karma starting level is 20
500: Karma gains 1 skill level (Total: 3/5)
600: Karma gains -1 cost (Total: -3)
700: Karma changes classes from Vampire Princess(Platinum) to Immortal Princess(Black)
800: Karma gains 1 skill level (Total: 4/5)
900: Karma gains -1 cost (Total: -4)
1000: Karma gains 1 skill level (Total: 5/5)
1200: Karma gains -1 cost (Total: -5)