I have two Nightmare Shepherds and one Kird Ape in play and my opponent sends Kird Ape to the graveyard, does this trigger both Nightmare Shepherds or just one?
And my second question is, if I have one Nightmare Shepherd in play and three Kird Apes and my opponent casts Wrath of God, does Nightmare Shepherd's ability still trigger and I create three 1/1 tokens?
For your first question: The triggered ability of each Nightmare Shepherd will trigger (C.R. 603.2), assuming Kird Ape is a nontoken creature. However, in general at most one of those abilities will do anything useful. This is because if you exile Kird Ape due to one Nightmare Shepherd ability, you won't be allowed to exile Kird Ape with the other, even if it returns to the graveyard before the other ability resolves, since Kird Ape has already left the graveyard (C.R. 400.7, 400.7d, 400.2, 608.2d). See also this thread.
For your second question: Nightmare Shepherd's triggered ability will trigger each time a "nontoken creature you control [other than Nightmare Shepherd] dies", even if Nightmare Shepherd dies at the same time as that other creature (C.R. 700.4, 603.6c, 603.10, 603.10; see also this thread). In this case, once Wrath of God destroys all creatures, the ability will trigger three times assuming all three Kird Apes are nontoken creatures and no other creatures are on the battlefield. Assuming those cards are still in the graveyard when each of those abilities resolves, you may exile the corresponding card and create a token if you do (for a total of up to three tokens).
EDIT: Add assumption after comment 4 was posted. Further edited after comment 5 was posted.
EDIT (Dec. 22, 2021): Note that the game has no notion of "board wipes" as such. There are numerous spells that can negatively affect all permanents of a certain kind on the battlefield at the same time — Wrath of God, Blasphemous Act, Armageddon, Obliterate, Upheaval, Decree of Annihilation, etc. — and they don't all affect the permanents they apply to in the same way.
I have two Nightmare Shepherds and one Kird Ape in play and my opponent sends Kird Ape to the graveyard, does this trigger both Nightmare Shepherds or just one?
It triggers both, but only one will be able to give you a token Kird Ape, because of the "you may exile it. If you do..." clause. The triggers go on the stack separately. As the first resolves, you need to exile the Kird Ape card from your graveyard to get a token. Then, when the second resolves, you can't exile the Kird Ape card which is no longer in your graveyard, and you get a token only if you do, so you don't.
And my second question is, if I have one Nightmare Shepherd in play and three Kird Apes and my opponent casts Wrath of God, does Nightmare Shepherd's ability still trigger and I create three 1/1 tokens?
Yes. Die triggers and other leave-the-battlefield triggered abilities trigger based on their existence prior to the trigger event (the move from battlefield to graveyard), so the Shepherd triggers for creatures that die at the same time as itself.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
I have two Nightmare Shepherds and one Kird Ape in play and my opponent sends Kird Ape to the graveyard, does this trigger both Nightmare Shepherds or just one?
Both will technically trigger, but only one ability will actually do anything. If you exile the Kird Ape for the first ability that resolves, that option will not exist for the second resolution.
118.12. Some spells, activated abilities, and triggered abilities read, “[Do something]. If [a player]
[does, doesn’t, or can’t], [effect].” or “[A player] may [do something]. If [that player] [does,
doesn’t, or can’t], [effect].” The action [do something] is a cost, paid when the spell or ability
resolves. The “If [a player] [does, doesn’t, or can’t]” clause checks whether the player chose to pay
an optional cost or started to pay a mandatory cost, regardless of what events actually occurred.
Example: You control Standstill, an enchantment that says “When a player casts a spell,
sacrifice Standstill. If you do, each of that player’s opponents draws three cards.” A spell is
cast, causing Standstill’s ability to trigger. Then an ability is activated that exiles Standstill.
When Standstill’s ability resolves, you’re unable to pay the “sacrifice Standstill” cost. No
player will draw cards.
Example: Your opponent has cast Gather Specimens, a spell that says “If a creature would
enter the battlefield under an opponent’s control this turn, it enters the battlefield under
your control instead.” You control a face-down Dermoplasm, a creature with morph that
says “When Dermoplasm is turned face up, you may put a creature card with morph from
your hand onto the battlefield face up. If you do, return Dermoplasm to its owner’s hand.”
You turn Dermoplasm face up, and you choose to put a creature card with morph from your
hand onto the battlefield. Due to Gather Specimens, it enters the battlefield under your
opponent’s control instead of yours. However, since you chose to pay the cost, Dermoplasm
is still returned to its owner’s hand.
608.2d If an effect of a spell or ability offers any choices other than choices already made as part of
casting the spell, activating the ability, or otherwise putting the spell or ability on the stack, the
player announces these while applying the effect. The player can’t choose an option that’s
illegal or impossible, with the exception that having a library with no cards in it doesn’t make
drawing a card an impossible action (see rule 121.3). If an effect divides or distributes
something, such as damage or counters, as a player chooses among any number of untargeted
players and/or objects, the player chooses the amount and division such that each chosen player
or object receives at least one of whatever is being divided. (Note that if an effect divides or
distributes something, such as damage or counters, as a player chooses among some number of
target objects and/or players, the amount and division were determined as the spell or ability
was put onto the stack rather than at this time; see rule 601.2d.)
Example: A spell’s instruction reads, “You may sacrifice a creature. If you don’t, you
lose 4 life.” A player who controls no creatures can’t choose the sacrifice option.
And my second question is, if I have one Nightmare Shepherd in play and three Kird Apes and my opponent casts Wrath of God, does Nightmare Shepherd's ability still trigger and I create three 1/1 tokens?
It triggers three times, and you can create three tokens that way. (You don't have to.)
603.10. Normally, objects that exist immediately after an event are checked to see if the event matched
any trigger conditions, and continuous effects that exist at that time are used to determine what the
trigger conditions are and what the objects involved in the event look like. However, some triggered
abilities are exceptions to this rule; the game “looks back in time” to determine if those abilities
trigger, using the existence of those abilities and the appearance of objects immediately prior to the
event. The list of exceptions is as follows:
603.10a Some zone-change triggers look back in time. These are leaves-the-battlefield abilities,
abilities that trigger when a card leaves a graveyard, and abilities that trigger when an object
that all players can see is put into a hand or library.
Example: Two creatures are on the battlefield along with an artifact that has the ability
“Whenever a creature dies, you gain 1 life.” Someone casts a spell that destroys all
artifacts, creatures, and enchantments. The artifact’s ability triggers twice, even though
the artifact goes to its owner’s graveyard at the same time as the creatures.
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Why bother with mere rulings when so many answers can be found in the Rules?
And my second question is, if I have one Nightmare Shepherd in play and three Kird Apes and my opponent casts Wrath of God, does Nightmare Shepherd's ability still trigger and I create three 1/1 tokens?
For your second question: Nightmare Shepherd's triggered ability will trigger each time a "nontoken creature you control [other than Nightmare Shepherd] dies", even if Nightmare Shepherd dies at the same time as that other creature (C.R. 700.4, 603.6c, 603.10, 603.10; see also this thread). In this case, once Wrath of God destroys all creatures, the ability will trigger three times assuming all three Kird Apes are nontoken creatures and no other creatures are on the battlefield. Assuming those cards are still in the graveyard when each of those abilities resolves, you may exile the corresponding card and create a token if you do (for a total of up to three tokens).
EDIT: Add assumption after comment 4 was posted. Further edited after comment 5 was posted.
EDIT (Dec. 22, 2021): Note that the game has no notion of "board wipes" as such. There are numerous spells that can negatively affect all permanents of a certain kind on the battlefield at the same time — Wrath of God, Blasphemous Act, Armageddon, Obliterate, Upheaval, Decree of Annihilation, etc. — and they don't all affect the permanents they apply to in the same way.
It triggers three times, and you can create three tokens that way. (You don't have to.)