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You can't activate Selesnya Eulogist's ability without a target creature card in a graveyard.
If the target card is an illegal target by the time Selesnya Eulogist's ability tries to resolve, the ability doesn't resolve. You won't populate.
Any enters-the-battlefield abilities of the copied token will trigger when the new token enters the battlefield. Any "as [this creature] enters the battlefield" or "[this creature] enters the battlefield with" abilities of the copied token will also work.
The new token doesn't copy whether the original token is tapped or untapped, whether it has any counters on it or Auras and Equipment attached to it, or any noncopy effects that have changed its power, toughness, color, and so on.
Populate doesn't target the creature token you're copying. You choose that creature token as you're taking the populate action. You can choose any creature token you control. If a spell or ability causes you to create a creature token and then instructs you to populate, you may choose to copy the token you just created, or you may choose to copy another creature token you control.
If you control no creature tokens when you populate, nothing will happen.
If you choose to copy a creature token that's a copy of another creature, the new creature token will copy the characteristics of whatever the original token is copying.
The new creature token copies the characteristics of the original token as stated by the effect that created the original token.
