A question that's very difficult to answer (esp. as you open up different questions). The short answers would be: No, not all Mexicans consider themselves to be indigenous people. And yes, there has been racial mixing (oftentimes forced) since the beginning of colonisation in Mexico, as in other Spanish colonies. Let's see for a longer answer.

My background is more in colonial history, so I'll focus mostly on these and less on current perceptions, but will try to bring it up to modern times. First a short intro on native people in Mexico: The category of native or "indio" only came into being with Spanish "conquest". Before then but also in part until now, various population groups had different identities, often based on common history and certain localities. You might've heard of different groups like the Maya or the Aztecs, and there were and are many other indigenous groups, often with their own languages and cultures. According to a national census from 2015, 21,5% or ca. 25 million of Mexico's population self-identify as being indigenous, of various ethnic groups (census, in Spanish).

My focus on the following will be more on central Mexico and the Nahua (later known as Aztecs). The Nahua in turn were not one homogenous group, but were (at the time of Spanish contact) dominated by the Mexica, who had different groups as allies, but also enemy groups within the territory they nominally controlled. Due to the complex (and somewhat controverse) topic I'll not try and give an impossible overview, but rather focus on three thematic groups, and try and tie it together in the end.

I. The casta system

This system was brought by the Spaniards to their overseas possessions. It had roots in medieval precedents, the "purity of blood" degrees used in Christian Iberia to distinguish between Christians and (converted) Muslims and Jews. Simply put, in the Americas the casta system was supposed to hierarchically distinguish between different social groups. María Martínez has argued that "in Spanish America, the notion of purity gradually came to be equated with Spanish ancestry, with “Spanishness”, an idea that had little significance with the metropolitan context. The language of blood and lineage also underwent modifications. Nonetheless, at the end of the colonial period, the concept of limpieza de sangre was still partly defined in religious terms”.

Casta groups included the Spanish at the top of the pyramid, who held the highest posts including that of Viceroy; and "indios" as mentioned, who were under the Crown's protection if they converted, and if noble could continue to hold a certain influence. Others were the "creoles" or Europeans born overseas; "mestizos" of mixed European-native origin; people of African origin; and Asians (often from the Philippines). Then we have a large variety of casta due to mixing between the groups (hopefully answering your last question). The important thing to understand here is that this was an ideal system which did not actually work completely as envisaged by the Spanish. The bizarre nature and the crazy amount of subdivisions in this "ideal type" system between groups becomes (hopefully) clearer in this casta painting. Such painting became current esp. in the 18th c., and were sometimes even commissioned by kings and viceroys, as in colonial Peru.

So we have at least in earlier colonial times (ca pre 17th c.) still quite of lot of flexibility between these groups. We know of noble Indians and mestizos who described themselves and were perceived as "Spanish". On the other hand, noble Spaniards starting with conquistadors tried to marry into the native nobility in order to gain legitimacy and privileges. R.D. Cope has spoken of "the limits of racial domination" in this context, meaning that even commoners of lower castas had a certain agency and could swith between groups. Nonetheless, he argues that there were long-term effects to the casta system and its exertion of elite Spanish control; and that it would influence people's perceptions of these groups.

II. "Creoles" and "indios"/ a neo-Aztec identity

Now that we've got the "basics" covered we can look to the very beginnings of the modern Mexican national identity, one that is infused with Aztec/Mexica symbolism, heroes and mythology. Where did this connection originate? D.A. Brading has added to the concept of "creole patriotism" for Spanish America. For Mexico, he describes how certain creole authors, starting in the 17th c., would draw on writing of indigenous authors to create a patriotic identity distinct from the Spaniards. This had to do with the creoles holding much influence and economic power by then, but not being able to attain the highest posts and prestige in colonial society.

Brading also notes that creole scholars like Carlos Siguenza y Góngora and Javier Clavijero held the native and esp. the Aztec past and civilization in the highest regard, in order to contrast/compare it with European civilisations. Then again, these scholars held negative views (and sometimes even contempts) for native commoners in their own times.

In a more recent study, P. Villela has traced more connections between native and creole scholars. According to him, there was actually much exchange between these groups. This meant especially a narrative developed by native authors soon after conquest, which highlighted their ancestors legacies. At the same time, it did not pay much attention to the conquest, which was seen as rather one more event in their timeline. Similarly, creoles would also narratively erase the conquest, if for other reasons.

I note all these connections because they form a basis as mentioned of a national narrative and identity for Mexico, that would take shape during the 19th century especially after inependence. Its modern variant draws on developements after the Mexican revolution. This influential narrative has meant, as mentioned: 1) To describe pre-hispanic native civilizations, esp. the Aztecs, as a main constituent of the Mexican nation and its culture. 2) To downplay the importance (and violent nature) of "conquest" . 3) As with the earlier creole writers, modern day (not pre-conquest) native people are often seen in a much less positive/idealized light, and are still being discriminated against.