There are three Poké Balls with additional effects (other than catch rate modifiers).
Friend Ball: Sets caught Pokémon's friendship to 200.
Heal Ball: Fully restores a caught Pokémon's HP, PP, and status.
Luxury Ball: Doubles the rate in which the contained Pokémon receives friendship.
To answer your question about Eggs:
A Pokémon hatched from an Egg starts with 120 happiness, and it goes
on a scale of 0-255. Caught Pokémon start with a species-specific base
friendship stat, and Pokémon caught in a Friend Ball start with 200.
Yes, as the Poké Balls can be inherited starting in X/Y, a Pokémon can
inherit the boosting effects of a Luxury Ball even if it was hatched
from an Egg, as long as it inherited the Luxury Ball.