The question of dead pigeons is addressed in a current post at Atlantic Cities. Short answer: dead pigeons are tasty treats for many predators.
And baby pigeons? A Straight Dope reader offered a reprise of a National Lampoon riff:
Re your recent comment on baby pigeons: don’t be fooled by false sightings from gullible bird lovers. The blunt truth is this: the pigeons you see all over the city are the baby pigeons. The adult has a wingspan of 8-12 feet. When they reach adulthood they fly to remote mountain fastnesses and live off the occasional tourist. I do not, however, subscribe to the theory that the adults will one day return en masse to wreak vengeance on us a la The Birds.
— Bob W., Melrose Park, Illinois