[RPG] create a character that’s above the PHB-given height and weight restrictions

This is a question that is probably more about the world your campaign is located.

Dragonlance and Forgotten Realms (in numerous novels) indicate that elves at the age of 20 would appear to our eyes like human children. They grow slowly, and learn slowly. Time passes for them subjectively different then quick-paced humans.

The Player's Handbook indicates "elves reach physical maturity at about the same age as humans," but this is not a universal conception for elves. Going with that thought though, the implication is they don't reach emotional/mental maturity at that age.

The fragile 20 year old elf may suffer all kinds of malnutrition and loss of connection from being apart from its family and homelands. Traditionally, the magic of the Elven homelands and interaction is like nourishment. The 60-100 year old elf would appear and act similar to your 20 year old characters.

You are free to make your elves however you like in your world. For instance, in the harsh climate of Athas, where to the elves the wastes are considered Elven lands (multiple editions), the starting ages for elves are 20-ish. The destructive magical cataclysm of Athas had transformed Elves significantly, forcing them to mature and grow up faster (over thousands of years).

If the question is the rules as written in the Players Handbook, the starting age is listed there:

An elf typically c1aims adulthood and an adult name around the age of 100 and can live to be 750 years old.

This is probably based more or less on Tolkien sources, adapted for game purposes (In Tolkien Elves were immortal, but suffered over thousands of years when separated from the Undying Lands).

You can choose any age for your character, which could provide an explanation for some of your ability scores.

As to would it be weird, I give just one of a million possibilities: The elven character was plane-shifted from Athas (chose Outlander background); use these stats (no minuses in 5e, do +1 Int as per High Elf, convert everything else):
http://whinehurst.com/darksun/index.php?title=Races. [Wayback mirror]

Again, whether that is weird, also depends on your gaming world.

So for multiple reasons, the Player's handbook is giving one some artistic license here, with which a player can work with their DM and his/her game world.

Speaking from my experience here, there are a couple of things I would recommend:

• DM Cheat sheet - this was really helpful for me, since it gave me an idea of the basics of the character. I made sure the player created and provided me with this sheet at every level, not only to keep me informed with what their character was doing, but also to give me insight into what they thought was important about their character. This helped me to build obstacles that catered to their character's specific role, or at the very least what the PC perceived their role to be in the party. I would limit this cheat sheet to 1/2 a page at lower levels, and 1 page at higher levels, to ensure it remains concise and useful to the DM.
• Time Limit on turns - this was a more controversial move in our group, but definitely helped keep my PCs on top of their characters. If I as a DM had a question, or asked their character to do something, the complicated PC would have no more than a minute (or 2 depending on how complex the request was) to figure out the answer. By having the PCs aware that this time limit was in effect, it incentivized them to maintain a deep understanding of their character. Our table also had a high level of trust, so I wasn't too worried about my PCs fudging their characters in these time sensitive scenarios.
• In advance verification - this was something I only used once, but can still highly recommend. Before allowing my PC to create a complex, multi-classed character, I sat down with them and discussed why they wanted to create such a character, and how we could work together to streamline the process in-game (this was incidentally how we landed on the cheat sheet solution above). By having this talk in advance, you can understand why the player wants a convoluted character, and brainstorm alternatives if possible (eg Player: I want to be a healer, but also be able to fight. Me: well instead of multi-classing and trying to use 3 prestige classes, you could just main as a Cleric, and by taking feats X, Y, and Z still achieve this goal). If alternatives are not an option (eg Player: No, I really want to play [insert convoluted character]!) then at least you've now gotten some perspective on where they are coming from, why they have chosen this particular build, and laid the groundwork for future discussions should things go off the rail.

Introducing a metagaming requirement is an interesting idea, and something I had not considered. While this would be useful in the context of the game's lore and the character's development, if you are more concerned about how to deal with complicated characters as a DM, I would just warn you that complicated backstories can lead to complex character motivations, which in my experience, often leads into My Guy problems.

That being said, the flip side is that you are maximizing the number of plot hook you as a DM have available. Is the character adamant on being a bard/druid/green whisperer/sublime chord/arcane hierophant? Fine, but their complex background has made them a target of a group of purist druids, who believe their magics are being abused by an outsider for their own personal gain.

Ultimately, requiring as comprehensive a backstory as the character is complicated can work both ways, but independent of this, I would advise you to consider some of my suggestions listed above. They can definitely streamline the DM process, and make your life a lot easier.