Ozai declaring himself ‘Phoenix King’ is supposed to show how far around the bent Ozai has gone. And my it is far.
But if you view it in-universe (the show is set in a sino-centric universe) it becomes even weirder.
Let me explain:
The phoenix is the embodiment of female power i.e. the Empress; not male i.e. the Emperor. That would be the dragon.
The phoenix represents yin, the dragon yang.
So Phoenix King is kinda an oxymoron.
What he did is either declaring that he now is Emperor and Empress in one person. Ozai is taking on the yin part of the yin and yang, on top of the yang of ‘king’. He is now, by declaring himself phoenix and dragon, ‘complete’. Or, and I am not going down that creepy road, one could argue that by making Azula Fire Lord, he is making her the dragon/husband.
Now I understand that they wanted to run with the ‘reborn from ashes’ aspect, but it makes Ozai, in cultural and show context, sound even more insane than he is.
On a side note: There is a real phoenix crown, traditionally worn by the Empress (compare picture), and there are even similarities between Ozai’s helmet/crown and the real thing e.g. the wings on the sides.
Traditional Chinese art views red, black, blue-green, yellow and white as the base colors. Each color corresponds with an element (note: NOT the Greek elements) Water, fire, wood, metal and earth. In the past the Theory of the Five Elements was used to select colors. More.
The Red Dragon
Red is the color of prosperity, good fortune and happiness. Red dragons are associated with luck, fire and passion. Red dragons are the dragons of summer and the South. Red also stands for happiness.
The Blue Dragon
Blue/Green is the color of nature, serenity and growth. Blue dragons are the dragons of spring and the East. Blue also symbolizes immortality.
Note: Blue and Green are seen as the same color, not the same shade.
Now, why is Azula a blue/green dragon in Zuko’s dream? It’s a pun. Again :P Take a look at the table: Green - East - Azure Dragon
Remember Avatar Roku’s temple in Season one, how the sun hit the statue on the Winter Solstice? Now compare the floor pattern and the door/sun opening mechanism in the Sun Warrior temple. They match.
On the Sun Warriors’ island Zuko tricks the door into opening, it was not yet time for it to do so. When Zuko and Aang do the dragon dance/imitate the ceremony, the egg rises, but since they’re too early, it does not hatch.
What this implies is: On the next solstice, when the sun hits the red gem and opens the door, the sun warriors will gather for the ceremonial dance. The egg will rise from the floor and the dragon will hatch.
Note: Imperial dragon have FIVE claws, lesser dragons only three or four.
Other recipes include:
belly of a clam
scales of a carp,
wings of a bat,
and claws those of an eagle
Chinese dragons are associated with water, not fire:
Initially, the dragon was benevolent but the Buddhists introduced the concept of malevolent influence among some dragons. Just as water destroys, they said, so can some dragons destroy via floods, tidal waves and storms. They suggested that some of the worst floods were believed to have been the result of a mortal upsetting a dragon.
Chinese dragons are occasionally depicted with bat-like wings growing out of the front limbs, but most do not have wings, as their ability to fly (and control rain/water, etc.) are mystical and not seen as a result of their physical attributes.