I've noticed a trend towards 10-foot squares in 5e so most dragons should fit in the dungeon just fine. Even with 5-foot squares a dragon can squeeze, and it sucks for them to fight in those conditions, but most dragons would just be like, "eh, I'm a damn dragon, I can take it." The only problem would be 2.5-foot wide passages meant for humanoids to squeeze through, but I see those very rarely in actual adventures. And the Apex Predator who turns Huge and becomes 15x15 might have to squeeze a lot more and would not be able to go down 5-foot passages, but by 17th level there should be ways to work around that, like a friend with teleportation spells, or shrinking spells, or taking the Change Shape feat.
Here are my thoughts on game balance:
- At low levels, they are clearly overpowered. The racial traits alone are disgustingly good. I just couldn't figure out how to balance flight and non-humanoid type without making dragons seem artificially gimped.
- By upper levels, I think it evens out. When everybody in the party is attacking 4 times per round, flying, slinging meteor swarm, doing 180 points of damage with Assassinate, etc., then having claws and breath weapons and 26 Strength doesn't seem as impressive.
- The Maximum Strength ability is deceptive. It's meant to a) compensate for lack of magic weapons/armor, b) compensate for lack of Fighting Style, Rage, Smite, hunter's mark, etc., and c) siphon off Ability Score Improvements and feat selections by encouraging you to spend them on Strength. So in a way dragons get fewer ASI than other classes because they need to keep bumping their prime stat well past 8th level in order to take maximum advantage of an important class feature.
- Likewise, the natural weapons (2d6) are no better than a greatsword, and most other abilities are impressive-looking but mediocre once you do the math and compare it to what a dedicated warrior or spellcaster would do. Dragons look tough but they are secretly a hybrid class, sacrificing martial prowes for spell-like breath weapon damage.
- The archetypes are a big question mark, balance-wise. I agonized a lot over things like spellcasting progression and the damage of things like cataclysm and tail slap, and in the end, I went with whatever looked like about the same amount of damage as other classes were dishing out, but it is not well-vetted.
If I were going to use dragons in a mixed group with ordinary characters I probably wouldn't allow them until 5th level. At that point, most classes have some cool stuff going on, and while Fly 60 feet is still a great ability, it's no longer game-breaking.
An all-dragon party would be super sweet. That is where the multiclassing rules come in. I could totally see someone taking a few levels of dragon and then branching out into fighter, paladin, or sorcerer. Dragon warlock would be fantastic. It would really be a different approach to D&D fantasy than your standard murder-hobo campaign.