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.
You can use Animal Handling only to ride the dragon
You can use Animal Handling by the rules to interact with a Dragon, if the dragon is willing to serve as your mount.
Here is Animal Handling (p. 178, PHB):
Animal Handling. 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. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.
The examples given are not neccesarily an exhaustive list:
- calming down domesticated animals
- intuit an animal's intentions (domesticated or not)
- controlling a mount in a risky maneuver or keep it from being spooked (not limited to animals at all)
What are mounts?
Mounts ar listed in the Mounts and Other Animals table and include camels, donkey or mules, elephants, horses, ponys, and mastiffs. The rules further state:
Mounts other than those listed here are available in the worlds of D&D, but they are rare and not normally available for purchase. These include flying mounts (pegasi, griffons, hippogriffs, and similar animals1) and even aquatic mounts (giant sea horses, for example). Acquiring such a mount often means securing an egg and raising the creature yourself, making a bargain with a powerful entity, or negotiating with the mount itself.
So, essentially any creature you can ride could be a mount. And you can ride a dragon in D&D. Even a young dragon is Large size, and thus large enough to act as a mount for a medium-sized or small character.
The challenge is to get the dragon to agree to serve as your mount in the first place. For that you actually convince the dragon to act as your mount. Could you use the skill, if you interpret "calming" as generally influencing? Not by the examples in the skill, as the dragon is not an animal.
What is an animal in the game?
There is no explict definition of what counts as an animal in the game, so we need to use the dictionary. Here is Merriam-Webster's:
any of a kingdom (Animalia)]** of living things including many-celled organisms (...)
Kingdom animalia is a biology technical term. You can make the case that as it refers to real-world, naturally occuring animals listed thereunder, this defintion means that in the game, only such real-world animals should be covered.
Another approach is to look at monster types (MM, p. 6f). There are only 14 distinct monster types, and each creature in the game must have exactly one of them. For example, the Half-Dragon is just a humanoid, not a dragon humanoid. Animal is not one of the monster types, but animals are listed as a subtype of Beast:
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.
You can make the argument that this list is not exahustive, and other creature types like Monstrosity could be considered to include animals too. However, none of the other monster type descriptions includes any reference to animals. Dragons do not, and neither do Monstrosities:
Monstrosities are monsters in the strictest sense -- frightening creatures that are not ordinary, not truly natural, and almost never benign. Some are the results of magical experimentation gone awry (such as owl bears), and others, are the product-of terrible curses (including minotaurs and yuan-ti). They defy categorization, and in some sense serve as a catch-all category for creatures that don't fit into any other type.
I think this description makes it unlikely that even monstrosities which by their looks could be mistaken as animals of some kind, like an Owlbear or a snake-transformed Yuan-Ti, are part of the animal category. They defy categorization.
For both these reasons, I'd rule that animals for the game are just natural animals, maybe including giant versions of those, and nothing else. But there remains a slight fuzziness, so I think this is an area each DM needs to decide, in the end.
What can you do with a skill
The game only has a small number of skills that you can be proficient in (18, in total). The DMG (on p. 239) advises the DM on handling proficiency as follows:
When you ask a player to make an ability check, consider whether a skill or tool proficiency might apply to it. (...) One way to think about this question is to consider whether a character could become better at a particular task through training and practice. If the answer is no, it's fine to say that no proficiency applies. But if the answer is yes, assign an appropriate skill or tool proficiency to reflect that training and practice.
And there is a lot of things that and you could try to do or learn to get better in, that is not covered by any of the skills. For this reason, many DMs interpret the skill descriptions generously, and allow you to do things using the "closest" skill if you can make a good case for it, to add your proficiency to the test.
Here however, I would doubt that Animal Handling is the right or closest skill when it comes to influencing: you are dealing with a dragon which firstly is not an animal, and secondly has an intelligence that is much higher than that of an animal. The right ability score to use here would likely by Charisma, and the best skill, Persuasion or Intimidation.
Your DM may be more generous, and allow you to use it, or maybe allow you to use its proficiency bonus on a Charisma check as per the Variant rule on page 175.
1 You could use this phrase to claim Celestials and Monstrosities must include animals, because Pegasi are Celestials, and Hippogriffs and Griffons are Monstrosities, and "similar animals" in the same enumeration implies that they count as animals, too.