Birds start having babies when they grow up. It's like a natural instinct. They lay eggs to keep their species going.
Birds have stages: hatchlings, nestlings, fledglings. Maturity varies; Common Quails lay at 6-8 weeks, Californian Condors wait 6-8 years.
No, birds don't exhibit true paedogenesis. Neoteny is seen in flightless birds like Ratites, not in egg-laying.
Flightless birds like ostriches have juvenile traits, but they reproduce sexually, not paedogenically.
Birds mostly lay eggs in spring due to warm weather. Some species nest multiple times yearly, even in winter.
Birds have a cloaca for waste, eggs, and sperm. Internal fertilization, yolk formation, and egg laying follow. Embryo develops in incubation.
Birds' egg-laying varies: quail at 1.5 months, hens at 4-8 months, Great Tits at 1 year, bearded vultures at 5-7 years.
Once the ovulation has been triggered, the egg will proceed from yolk to fully formed egg over the course of several days. It must inevitably come out of the the body – be laid – even if the bird has not bred.