Does Phoebe, Head of S.N.E.A.K.'s ability used by a copy make the copy retain the rules text beyond end of turn?
The effect created by Phoebe's ability will affect the object permanently (i.e. until the object ceases to exist by leaving the battlefield).


  611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.




Copying an object whose text was "stolen" will still copy its original text.


  706.2. When copying an object, the copy acquires the copiable values of the original object’s characteristics and, for an object on the stack, choices made when casting or activating it (mode, targets, the value of X, whether it was kicked, how it will affect multiple targets, and so on). The “copiable values” are the values derived from the text printed on the object (that text being name, mana cost, color indicator, card type, subtype, supertype, rules text, power, toughness, and/or loyalty), as modified by other copy effects, by its face-down status, and by “as . . . enters the battlefield” and “as . . . is turned face up” abilities that set power and toughness (and may also set additional characteristics). Other effects (including type-changing and text-changing effects), status, and counters are not copied.


(Emphasis mine)