If you damage an Invader/Dahan and then replace it with another Invader/Dahan, does the new piece enter the game damaged?

If you damage an Invader/Dahan and then replace it with another Invader/Dahan, does the new piece enter the game damaged?

Yes. (Rulebook p. 18) When you replace a piece with another piece, the new piece comes in with whatever Damage the original piece had.
For example, if you deal 2 Damage to a City and then replace it with a Town, that Town is immediately destroyed.
If you replace a damaged piece with two or more pieces (as when "Dissolve the Bonds of Kinship" replaces a City with two Explorers), the damage (and Strife tokens, if applicable) get divided between the new pieces as you choose.
Note that if a piece is replaced as part of being saved from destruction (as is done by the Transforming Aspect), then Damage is cleared before the piece is replaced, so there's no Damage to divide. (Strife tokens are still divided normally, if relevant.)