Abandoning a creature will put it in the Castle Ruins, where it can then be picked up by any other player (including the one who has abandoned the creature). The person abandoning the creature does not get anything in return.
It is not possible to check the lineage of a creature while it is in the Ruins.
Creatures picked up from the Castle Ruins will show 'Ruins' as their Biome.
Zenith creatures will revert to their Adult stage upon abandonment.
There is only one restriction on abandoning: eggs younger than 1 hour cannot be abandoned. However, this time limit can be avoided by hatching the creature (for instance by feeding it) and then abandoning it, as this restriction only applies to eggs.
There is no restriction on the stage at which creatures can be abandoned; all stages from egg to adult are abandonable.