Edit:
Sorry, no. Replacement effects for entering the battlefield are applied without taking into account what the card would look like on the battlefield. So the shapeshifter gains the copy ability of the Double too late.
No. Because if the Image enters at the same time, then the Shapeshifter is not yet on the battlefield when you have to choose a creature to copy. Replacement effects are applied before the actual event happens, so that a modified event happens instead. You can only copy what is already on the battlefield. So you could do this if the Image entered after the Shapeshifter, even if that happens during the same spell's or ability's resolution.
No. Volrath's Shapeshifter's effect giving it "the full text" of a certain card is not a replacement effect, but rather a continuous effect that modifies Volrath's Shapeshifter's text (C.R. 612.5). Like any other text-changing effect on a permanent, it "do[es]n't apply before the permanent is on the battlefield" (C.R. 603.6b), which is when replacement effects that modify how a permanent enters the battlefield apply (C.R. 400.6). And by the time Volrath's Shapeshifter is on the battlefield and has the full text of Body Double (and additional text), it's too late to apply the Body Double replacement effect to make Volrath's-Shapeshifter-turned-Body-Double become a copy of something else.
After rereading C.R. 614.12, I have changed the answer I gave in comment 3 as follows.
In the circumstance quoted in comment 3, with the assumption given in that comment: Yes.
Volrath's Shapeshifter's effect giving it "the full text" of a certain card (and additional text) is a continuous effect from a static ability. Here, this is a "continuous effec[t] from [Volrath's Shapeshifter's] own static abilit[y] that would apply to [Volrath's Shapeshifter] once it's on the battlefield" (because it would give Volrath's Shapeshifter "the full text" of Body Double, including Body Double's ability, and additional text once Volrath's Shapeshifter is on the battlefield), and this is one of the things that are "tak[en] into account" when determining "the characteristics of [Volrath's Shapeshifter] as it would exist on the battlefield" for the purposes of "determin[ing] which replacement effects apply and how they apply" (C.R. 614.12), and that includes the replacement effect from Body Double's ability as acquired by Volrath's Shapeshifter, in this scenario.
Therefore, Volrath's Shapeshifter would take into account that it would enter with "the full text" of Body Double (and additional text) in time to apply the replacement effect of Body Double's ability acquired this way to make Volrath's Shapeshifter become a copy of something else.
If Volrath's Shapeshifter and Phantasmal Image come into play at the same time, will Volrath's Shapeshifter be able to be a copy of Body Double of which I can choose the Protean Hulk as the creature to copy?
And then can I make Phantasmal Image a copy of Volrath's Shapeshifter which will copy the characteristics and also become a Protean Hulk?
Niv-Mizzet Reborn
Feather, the Redeemed
Estrid, the Masked
Teshar
Tymna/Ravos
Najeela, Blade-Blossom
Firesong & Sunspeaker
Zur the Enchanter
Lazav, the Multifarious
Ishai+Reyhan
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Yes.Edit:
Sorry, no. Replacement effects for entering the battlefield are applied without taking into account what the card would look like on the battlefield. So the shapeshifter gains the copy ability of the Double too late.
No. Because if the Image enters at the same time, then the Shapeshifter is not yet on the battlefield when you have to choose a creature to copy. Replacement effects are applied before the actual event happens, so that a modified event happens instead. You can only copy what is already on the battlefield. So you could do this if the Image entered after the Shapeshifter, even if that happens during the same spell's or ability's resolution.
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No. Volrath's Shapeshifter's effect giving it "the full text" of a certain card is not a replacement effect, but rather a continuous effect that modifies Volrath's Shapeshifter's text (C.R. 612.5). Like any other text-changing effect on a permanent, it "do[es]n't apply before the permanent is on the battlefield" (C.R. 603.6b), which is when replacement effects that modify how a permanent enters the battlefield apply (C.R. 400.6). And by the time Volrath's Shapeshifter is on the battlefield and has the full text of Body Double (and additional text), it's too late to apply the Body Double replacement effect to make Volrath's-Shapeshifter-turned-Body-Double become a copy of something else.EDIT (Jun. 6): Edited.
EDIT (Oct. 15): Correctness edit.
EDIT (Oct. 28): See comment 4.
In the circumstance quoted in comment 3, with the assumption given in that comment: Yes.
Volrath's Shapeshifter's effect giving it "the full text" of a certain card (and additional text) is a continuous effect from a static ability. Here, this is a "continuous effec[t] from [Volrath's Shapeshifter's] own static abilit[y] that would apply to [Volrath's Shapeshifter] once it's on the battlefield" (because it would give Volrath's Shapeshifter "the full text" of Body Double, including Body Double's ability, and additional text once Volrath's Shapeshifter is on the battlefield), and this is one of the things that are "tak[en] into account" when determining "the characteristics of [Volrath's Shapeshifter] as it would exist on the battlefield" for the purposes of "determin[ing] which replacement effects apply and how they apply" (C.R. 614.12), and that includes the replacement effect from Body Double's ability as acquired by Volrath's Shapeshifter, in this scenario.
Therefore, Volrath's Shapeshifter would take into account that it would enter with "the full text" of Body Double (and additional text) in time to apply the replacement effect of Body Double's ability acquired this way to make Volrath's Shapeshifter become a copy of something else.