Cards cannot be placed into the hand of a player who does not own them. Thus, if a player were to take control of their opponent's "Cheshire Cat, Guide to the Mysterious World" and immediately use the mode to "put a card from a removed area that was removed by this card into your hand," it would do nothing.
In order for this card to gain any Flying, First Strike, Swiftness and/or Precision ability, the removed card must have one or more of those abilities naturally and not from any effect or ability, be it a met condition, a paid cost etc.
Ex.: "Demon of the Crest, Namblot" can give Swiftness to this card because it originally has it, while "Red Riding Hood" cannot, even if her condition to gain Swiftness is met.