Class is #1 with a bullet. In fact, Levels (which at one time could have been called Ranks) referred only to Class level and were understood to be different measures when used for other factors in the game. Experience Points too were simply a matter of class. Two classes meant two XP totals. Class is #1 because it is the entire scope of the game for the player who is currently attempting to puzzle out how to perform it. The better one does, the more XP they receive. Is it the only means of gaining influence over the world? No, but your abilities within the class were specifically designed in relation to the world design to empower the player to explore that scope. A fighter could try exploring magic, but their magic use was far less than the magic-users. This gave the game niches and focus, while leaving enough overlap to avoid separate play.
Race, while it is definitely #3, does not beat out Ability Scores. Race defines the limits a PC can progress in a class. It also defines the limits of one's Ability Scores. AD&D included these A.S. minimums and maximums, but I'd suggest core playable races are limited to those which can be conveyed throughout the 3-18 ability scores. (The high end extraordinary %ile scores should probably be dropped in some cases). Non-core custom races are not always going to fit within the 3-18 spectrum. Those that don't overlap at all don't get to be playable races at all. Not many people want to play a rock with the INT of a rock, the DEX of a rock, or the movement of a rock. Is it a race? Sure, but it falls outside the spectrum of the game's scope. Demi-human races, heck, all races had all the definitions the default scope human class did, but they had their own unique abilities that fell outside the human scope. They add flavor to the game. However, as the rock example shows, they need for them to overlap the human default Ability Score range still places them in 3rd.
#2 Ability Scores. These define the scope of the game after Class. They are pragmatically set up to represent adult human abilities. This is because humans are assumed to be the players of the game. If you've ever played D&D with a rock, it's not quite the same experience as playing with another human I can tell you. I grant 3e raised the benefits of ability scores so high in relation to the rolls they influenced that making AS's matter more is a scary thought. The reason they were rolled earlier though was that they represented, like every roll in the game, something running through the structure of the game world. In the case of 3-18 they represent the adult human distributive pattern of ability. Everyone rolled these, even NPCs (but only when the score needed to be known for them). NPC races rolled differently for the distributions relevant to each one of those races. Tying AS's to multiple subsystems in the game allowed just six scores to have a widespread influence over every aspect of the game regardless of any class's scope. The problem in 3e was that the scores were turned into a default bonus & penalty spectrum uniform for every subsystem they were tied to (almost, there were some exceptions). Keeping them balanced in AD&D meant tying each other system to them on their own terms rather than a blanket effect.
#4 Customizable elements were largely tied to class, race, or ability scores. Alignment was a choice, but could be limited by class. Saving throws and Hit Points were predetermined by class. Movement so to by race. Languages were affected by all three, but were still choices. Coin was separate altogether as well as all equipment purchases, though Advanced D&D did differentiate here. Plus some classes required a items to really work functionally well like magic-users and spell books. All other customization was done through play. I know that to modern day eyes the game looks like a strait jacket in some respects with few choices after PC creation. But the subclasses are really the more limited if only because they were more highly focused. The thing is, a player defined their abilities through play and could effectively create their own subclass, which they could then train a youngster in to start. Yeah, that means adventuring your way into becoming a Paladin with those powers or a Ranger or anything F-M related really. Call it Feather Dancers or whatever with all those named maneuvers you made up along the way. These weren't bought and then used because the book said so. The player created them within the scope of their class, made of it what they wanted to be and do, and, with a little negotiation, the ref worked these into the system behind the screen. Drawing these #4 elements out into the rules in front the screen is what strait jackets customization IMO. It's not that that they are unimportant, but that they remain within 1-3's scope of the game so we don't get PCs who are rocks "with stripes!"
Lastly, character and characterization are largely irrelevant to role playing. That may no longer be common usage for the term, but the game was originally based upon definitions stemming from the 40s to the 70s. The idea was, we are addressing an imagined reality in the brain of another person. We can explore it without changing our personalities. In fact, in a lot of D&D play most exploration is referred to with I and We. It is me and us who are in the imagined situation at hand, not someone else. As a thinking game we discern how to accomplish our goals. The question, "What would someone else do who isn't me?" is not only not required for role playing, it goes against the original definition. That doesn't mean different definitions aren't applicable or fun, only that for understanding the intent and design behind D&D we are better off using the sole one in place when the game was created.
