What is Western? There are a lot of creatures that have become popular in more modern visions of 'western myth'. My answers intend to cover apex predators that are seen in a lot of modern high-fantasy that tends toward a western mindset, but originated elsewhere in a more refined form. Due to this, I focus on Classic Fantasy more than Western. These creatures appear in westernized classic fantasy, but the mythology stems from elsewhere and they are viewed differently in different areas of the world.
I am bundling a few types of great birds together as a single response because they are similar in many ways. I will discuss the Phoenix and Roc because they have a lot of lore behind them, but this could easily apply to other elements by altering them slightly, thus giving them the same elemental versatility that D&D style dragons do (or avoided to skip that trope).
Though considered primarily in modern context as large birds of prey and not denoted to a particular level of intelligence, in ancient societies they were a symbol of prosperity and good rule, only appearing for good and virtuous leaders. Its immortality makes it a special point, as reasoning may be the only way to deal with it.
In the Eastern Asian belief systems it was typically considered the equal and opposite pair to the dragon. It's appearance was considered fortuitous and associated with great rulers.
They certainly qualify as an Apex-Creature, and although modern western thought doesn't tend to give it as much credence for intellect, the myths about them exist throughout Europe and Asia, and many regard them as intelligent, even if they don't have automatic communication methods the way Dragons do in every myth. This can be simulated with either giving it normal speech capabilities, or a telepathic communication method.
If you make it magical enough (which is easy with their lore), it could easily shape-shift in to a human, most likely a female (as they represent the "Empress"), and allow it to communicate and manipulate through that the same way many Dragons are portrayed.
So to sum up the criteria:
Yes, they exist in both western and eastern fantasy, but are looked at as far more intelligent in eastern fantasy.
No. They are winged
Mythical enemies of dragons
It doesn't even need a mate to breed
According to a lot of lore out there (though not western lore so much) yes.
Control an Empire
Heavily associated with rulers, and could be seen as a way of determining rulership, should they decide to, thus manipulating a culture by their timely arrival.
As an extension of the Phoenix, I will offer another 'elemental great bird', the Roc.
In some versions of Sinbad's 1001 Nights, the Roc is considered to be the master of Aladdin's Djinn. It isn't considered to be intelligent, generally speaking, but it is an aspect of the element of Air, and thus associated as a supreme representation of what the Djinn were.
The Native American Thunderbird is considered a corollary and are considered to be intelligent, strong, and wrathful.
They weren't always considered as magical as the Phoenix, but could simply be seen as the Air equivalent to the Phoenix's fire, thus making them two sides of the same coin.
An interesting real-life corollary to the Roc is the Haast's Eagle from New Zealand/Australia. We know that it lived while humans did and we drove them to extinction. They had over 9 foot (almost 10 foot) wingspan, and that wingspan was considered small for the bird's size. It could be the source of legends of the Roc. They were able to take down the moa, which weighed about 500 pounds.