the former depends on the dragon. Some dragons especially ones with more snakelike scales, do molt.
As for the latter?
e g g 🥚
note however, that roughly 10% of dragons didn't hatch from eggs.
... they were former humans who turned into dragons.
(though one dragon scholar famously proposed that "humans are a kind of egg sometimes if you think about it", which prompted a loud shouting-and-roaring match at a council of dragon scholars)
