1) Eggs will appear on a weekly basis, just so certain trainers aren't swamped with a million eggs by forgetting about their Pokemon in the daycare for a month or longer.
2) That depends on the Pokemon's egg cycle! Take the number of egg cycles a Pokemon has, divide it by two, and that'll be the number of days it'll take to hatch.
3) There is no upper limit, but the daycare centre employees might get really annoyed and upset if they start to pile up.
4) Yes, trainers will be notifed by Pidgey Mail Express whenever there is an egg! Come and pick up your hellspawns before they overcrowd the building :/