Nahuatl was the language spoken by the Aztecs. According to my source (see below), the Aztecs actually called themselves the Mexica. So, yes, the whole country is named for them. We call them Aztec after Aztlan, their name for their original homeland.
Here are some vocabulary words gleaned from my source. Most of them are names of places, rulers or gods:
tlalli = earth
Huitzilopochtli = the hummingbird of the left [hand]
Huitzilhuitl = hummingbird feather
Quetzalcoatl = green-feather serpent
Coatepec = hill of the serpent
Chapultepec = hill of the locust
Coatlicue = Huitzilpochtli’s mother (I don’t know the meaning of her name, but apparently some kind of serpent.)
Xiuhcoatl = serpent of fire (a weapon)
Itzcoatl = serpent of obsidian
Cihuacoatl = woman snake (title for an advisor to the ruler)
coatepantli = serpent wall
Chicomecoatl = seven serpent (i.e., corn)
Chicomoztoc = seven caves
Therefore:
chicom- = seven (might have an underspecified vowel as its final sound)
coatl = serpent (l drops when followed by e)
huitzil = hummingbird
tepec = hill
Source: The Aztecs, by Brian M. Fagan, W.H. Freemand and Company, New York, 1984
Out of Babel Books 
It seems that the serpent holds an important place in their society – I’m especially intrigued by Cihuacoatl being the title for an advisor to the ruler, I wonder how that came about! Did your source say anything about that?
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Yes, there is a lot of mythology that I didn’t get into, and also a lot of mythology and early history that was lost. Serpents are a big deal. They’re on their temples, steps, walls, sometimes shown eating people with body parts strewn around. All the origin myths seem to involve a lot of violent dismembering, even by the usual standards of origin myths.
Quetzalcoatl, an important god/culture bringing figure, is the Feathered Serpent and he was worshipped by most of the cultures in the area under different names. (I have my own theories about that.) I think the Mayan name for him was Kukulkan.
There’s also a god called the Flayed Lord, shown wearing a human skin over his own skin. For his festival, they would flay the dead bodies of prisoners, and the priests would dance in the skins. I kinda think the Aztecs drew the short straw when it comes to gods.
Anyway, my guess about the “woman serpent” thing is that serpent indicates power, woman indicates an auxiliary position. But that’s just a guess. I didn’t get deep into it. It could be power + being the source of something.
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Ahhh that is fascinating! I don’t know much about Mayan culture, so thank you for sharing!
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