From when a dragon hatches from its egg to the end of its life, could a character use Animal Handling to interact favourably with it? If so, when (in terms of life stages of the dragon) would it apply and when does it cease to be effective (given that Dragons are highly intelligent and may not necessarily be counted as 'animals')?
To answer your question with a question:
Would you use Animal Handling on a human infant or a human toddler?
If your DM does allow use of Animal Handling, the time where that skill is applicable would be very short. Dragons are by nature very intelligent beasts. This possibly related discussion on Animal Handling is provided with a caveat: Animal Handling isn't the right tool for long, if at all.
Why is the Animal Handling skill a bad fit?
Dragons are smarter on average than humans are.
The age category "Young" will have an Intelligence better than that of the average human.
Example: 16 Int score for a Young Green Dragon. At "Young" a dragon already speaks the Common tongue. (Basic DM Rules, p. 52).
A Wyrmling (Green) has a 14 Intelligence, and speaks Draconic. (p. 95, MM, SSD assist appreciated).
Argument against using Animal Handling
Dragons are not Beasts in the way that the term is used in the game as a tag.
Dragons are large reptilian creatures of ancient origin and tremendous power. True dragons, including the good metallic dragons and the evil chromatic dragons, are highly intelligent and have innate magic.
Beasts are nonhumanoid creatures that are a natural part of the fantasy ecology. Some of them have magical powers, but most are unintelligent and lack any society or language. Beasts include all varieties of ordinary animals, dinosaurs, and giant versions of animals (Basic DM Rules, p. 2).
- Example Dragon (p. 52)
- Young Green Dragon: Large dragon, lawful evil
- Example Beasts (p. 9, 10)
- Allosaurus Large beast, unaligned
- Ape Medium beast, unaligned
As DM, I wouldn't allow it.
KorvinStarmast is correct, I think. The Animal Handling Proficiency is fairly tightly defined. On page 178 of the PHB is says, "When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked , or intuit an animal’s intentions, the DM might call for a Wisdom (Animal Handling) check."
Dragons (in general) aren't animals. They certainly aren't domesticated. They are thinking, self-aware creatures. Animal Handling is used for dealing with horses, cattle, dogs and other domesticated animals. I might allow someone to use Animal Handling to intuit whether a bear or wolf is about to attack, since the PHB says something about an animal's "intentions". That would be situational in my opinion though. The Proficiency has a very heavy implication that it is intended for animals you'd typically find in the company of humans either as mounts, pets, draft animals or whatnot.