If you are hatching an egg and you put it in your first slot whilst you were running around trying to hatch it, when it hatches would you have Pokemon that is already very happy?
No it can't.
"When a Pokémon is hatched from an Egg, its value is set to 120."
Although, eggs can apparently receive an increased Friendship upon hatching. "In Pokémon Diamond, Pearl, and Platinum, Eggs can be given a spa treatment at the Ribbon Syndicate building. This will cause the Egg to have an increased friendship upon hatching."
Hope I helped :)
No, it won't.
Just like the real world, Pokemon inside the egg won't see what you're doing to make it happy, so it won't raise the happiness.
Hope I helped!
No, it cannot. I've actually tested this before with two Larvesta, one whilst I walked around with while it was an egg, and one I just hatched. Both had a Friendship Value of 120 (which is the value all Pokemon hatched from eggs have).
Source: My test