Great questions! So the main group of Aztecs and other Indigenous people who came to Spain were slaves, with native slavery continuing until the 17th century. The more exceptional groups were some travelers and diplomatic voyages of elites to the Spanish court. Then there's the sole case of a descendant of Moctezuma's becoming a Spanish count.

• I've talked about most of this over here

• And you could also check out here on the Duques de Moctezuma

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For the second part: Officially the Spanish King was directly responsible for native Americans, as part of the Treaty of Tordesillas with the Pope. As I said diplomatic visits to the King happened but rarely. The Viceroy was "The Kings Living Image" and representative in New Spain, and we do have writings by Aztec nobles to the Viceroy.

But practically speaking the main representation was thought the city councils or Cabildos de Indios that existed in most larger Indigenous communities. By the 1570s there was also a court in Mexico City, the Juzgado de Indios, specifically for settling disputes for native people, e.g. often between communities and nobles.

Basically the Aztecs and other groups were strongly discriminated against in a hierarchical society ; at the same time there were various avenues for judicial representation which they used often and strategically - quite different from other European colonial powers at the time.

• If you excuse the serial linking : You may also be interested in this earlier answer of mine on some of this, on Aztec organisation and esp. the 2nd part on Tlaxcala.