Some more specific questions (feel free to address any or all, or some that I couldn't think of):
Would the people of native descent still call themselves Aztecs?
How well were they integrating into Spanish society?
What were the things they disliked most? Were there any things they liked better?
What did they understand or blame regarding disease?
How did they balance their religious views with the new Christian views?
It seems like the whole world changed for them relatively quickly, and I'm curious to know if they resisted it, accepted it, or faked compliance while nurturing their old ways of life.
It's difficult to give you an answer here because the question is so broad. I've written quite a bit on connected topics on here, some of which you might find interesting. They include literature recommendations, but let me know in case you're looking for readings on something more specific regarding this.
This one gives a more general introduction to political changes in early colonial times (not in Tenochtitlan though):
Here I'm looking at some noble families connected to Moctezuma II:
This one has a more general approach, and moves into current times:
Regarding some of your other questions I hope these aswers will be of help:
Would the people of native descent still call themselves Aztecs?
What did they understand or blame regarding disease?