The definition of Cosmic is incorrect, and Unique and Evasion are incomplete. Also, invulnerability is not optional (which the word can seems to imply).
If a character has Cosmic, it means that when they come into play (not just when recruited), they get a Cosmic counter. While they have that Cosmic Counter on them, they get the powers after the word Cosmic. If that character is stunned, they lose their Cosmic Counter. Also, not all Cosmic powers require removing the Cosmic counter to use them (actually, only a very few have that as part of their effect).
Unique is a characteristic that applies to characters, locations, and some equipment. If a character or equipment is Unique, then when it is recruited, if there is another character or equipment with the same name on your side of the field, that character that was on your side of the field is sent to the KO pile. Note that if the character or equipment just comes into play, but is not recruited, then uniqueness is not checked (an example of this would be Dr. Light's boost effect or Cyborg's boost effect).
Locations are similar, except that their uniqueness is checked when they are flipped. If, when they are flipped, there is another location with the same name in your resource row, the one or ones that were in your resource row will be sent to the KO pile (essentially costing you a resource). However, if you already have a location of a particular name, and you gain control of another face-up location with the same name, neither of the locations will KO, because no location with that name was just flipped.
Evasion is a payment power that reads: Stun this character -> At the start of the Recovery Phase, recover this character.
This recovery occurs before the recovery in the 'wrap-up' section of the recovery phase, and while it will usually mean that you will get to recover that character, because it is a payment power, it can be negated by an effect like that of Utility Belt.