Very developed.

Regional Market Towns

Were there regional market towns that were not administrative centers? Yes. Although many market towns were administrative centers - each “province” of the empire had a head town where tribute would be collected[1] - there were a few regional centers of trade that were not head towns. Not many, because the head town generally was picked as the head town because it was the largest and attracted the most trade. Examples of non administrative centers that were still regional trade centers include Huexotla[3], south of Tetzcoco in central Mexico, which featured products from as far away as the Yucatan Peninsula, and Tetellan[3], along the border with the Tarascan Empire, which traded with both Tarascan and Aztec towns. An argument could be that the city of Tlatelolco fits this criteria. Although technically a province[1], it was the only city in its province[1], and was located on the same island as Tenochtitlan. Being on the same island as Tenochtitlan, it would not have been used as a major administrative center, and its market far overshadowed any other function the city may have had. Tlatelolco’s market was “probably the largest market in Mesoamerica in the postclassic”[3], featuring utilitarian and exotic goods from across the empire and from beyond its borders.

Commercialized Agriculture

Commercialization means “primarily for profit”. Some crops were primarily for profit but others, primarily maize, were more for survival, and secondarily for profit. Since you asked about Tenochtitlan in particular, I can say that the farmers in Tenochtitlan were not exporting crops. Tenochtitlan was not a self-sufficient city [2], and even the intensive chinampas (artificial islands built in the lakes) could not feed the 212,500[4] people living in it before the conquest, it had to rely on additional food from the rest of the basin. (Agriculture in the city accounted for only 5% of the city’s subsistence needs according to Hassig.) The lake system that Tenochtitlan was built in was a large part of the reason it was able to grow so large, allowing the creation of chinampas and the expansion of its food sources beyond what was normally available to mesoamerican cities through canoe trade. This is illustrated by this chart. As you can see, the vastly increased carrying capacity of the canoe and the ability to go accross water expanded Tenochtitlan’s economic hinterland to an 18 mile radius around the entire lake system of the valley of mexico. According to Hassig “Modern data describe 7-meter and 10- to 15-meter canoes in the lakes of the Valley of Mexico, similar to those mentioned by Diaz del Castillo, in which a single man could pole over a ton of maize.”[2] (That gives you an idea of what the canoes were like. As for the numbers of canoes visiting Tenochtitlan on a daily basis, estimates vary from 200,000 to 70,000 to 50,000[2])

Normally, a city’s hinterland would have been limited to what tlamemes (essentially professional stuff carriers) could reach while the good was still commercially viable and could make a profit. Maize had a relatively low range, while goods like cotton and cacao beans were still economically viable at longer distances.[2] So, it really depends on the particular crop for the degree of commercialization.

Interregional Commerce

Definitely! The Aztec Empire had a thriving internal trade network, primarily driven by the merchant class, called “pochteca” in nahuatl. Moving goods themselves or employing tlamemes, these merchants would sell a variety of luxury goods at trade centers such as Tlatelolco, Tochtepec, and Xoconochco.[3] Luxuries included but were not limited to copper and bronze bells, gold jewelry, greenstone jewelry, feather ornaments, decorated clothing, fine salt, cacao, and painted manuscripts.[5]

Money

It is true that cacao beans were used as currency.[6] (Coins were not used, and were never produced) In fact, there were a few other types of currencies with generally agreed upon values as well, such as the T-shaped “copper axe money”, and large white cotton capes called quachtli.[6] Money was very widely used, especially at important marketplaces. Spanish historian Oviedo y Valdes said of cacao beans “They guard them and hold them in the same price and esteem as the Christians hold gold or coin, because these almonds are regarded as such by them, as they can buy all things with them.”[6] As for the government’s role in the use of money, I do not know, perhaps someone else can explain how the different types of money were distributed and whether or not the government had a role in determining the value.

Feeding Tenochtitlan

I already explained this a bit in the commercialized agriculture section, but in summary it was a mixture of both methods. The chinampas, tribute from conquered cities, larger than average economic hinterland, and people trying to make a profit on maize all feed the city and allowed it to reach its great size.

Occupations in Tenochtitlan

I do not know any exact population ratios for Tenochtitlan, or any other city. Everyone in your question (merchants, servants, slaves, artisans, and commoners) would have had residences in the city. Farmers would live next to their chinampas, and many of the elites, including the famed military orders (Jaguars and Eagles) would live near the royal palace.[7] You might find Hassig’s estimate of 1,612 to 2,150 males nobles ages 20 to 50 present in Tenochtitlan interesting.[7]

Other Big Cities

Tenochtitlan was definitely the largest city, but there were many other big ones. Examples include Cholula, famous for the Great Pyramid of Cholula, which had a population of 30,000 to 50,000. Tzintzuntzan, the capital of the Tarascan Empire, had a population of 25,000 to 35,000.[8] Tetzcoco is estimated at 24,000.[4] Note that these figures only include the city proper and not the surrounding agricultural areas. The population of the entire valley of mexico, combining all the cities and towns, is estimated to have been from 1,200,000 to 2,650,000.[7]

Interstate Trade

Pochteca merchants did trade in the trade centers of competing and even hostile states, but not all merchants were politically neutral. In fact, merchants were often employed as spies, especially by the Aztec Empire. Some merchants would disguise themselves as locals to gather information, and would be killed if it was found out that they were actually spies. The Aztecs used enemy assault of their merchants as a reason for war on multiple occasions.[7]

Back to your direct question, the Aztecs may have outlawed merchants of enemy states from trading in their cities[2], and the Aztecs would try to keep good relations with independent trade centers they had yet to conquer, such as Xicalango[3]. Trade between the Aztecs and the Tarascans was primarily funnelled through the border town of Tetellan [3] that I discussed earlier. For the mixtecs, there was a city state called Coaixtlahuaca which Aztec merchants regularly visited, until the ruler ordered Aztec merchants to be killed - promptly resulting in Aztec conquest and a lot more Aztec merchants than before.[3]

Sources

[1] The Essential Codex Mendoza - Edited by Francis F Berdan and Patricia Rieff Anawalt

[2] Trade, Tribute, and Transportation by Ross Hassig

Articles in The Postclassic Mesoamerican World - Edited by Francis F Berdan and Michael E. Smith

[3] International Trade Centers by Janine Gasco and Frances F. Berdan

[4] Aztec City-States in the Basin of Mexico and the Morelos by Michael E. Smith

[5] Key Commodities by Michael E. Smith

[6] And International Economy by Frances F. Berdan, Marilyn A. Masson, Janine Gasco, and Michael E. Smith

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[7] Aztec Warfare: Imperial Expansion and Political Control by Ross Hassig

[8] Tariacuri’s Legacy: The Prehispanic Tarascan State by Helen Pollard

Edit: Fixed nightmare formatting.

Since /u/DownvotingCorvo has given a superb answer already, I want to take a slightly different tack and discuss the interplay between trade and tribute.

The Aztecs are often thought to be a nation spurred on to conquest by religious fervor. While this is not entirely untrue, and Duran famously has Tlacaelel giving a speech declaring perpetual war on the Aztec's neighbors, who would serve as food for the hunger of the gods, it is not the whole picture either. Expansion of trade and securing both staple and luxury goods was equally, if not more important.

The economic impetus for war is not unique to the Aztecs in Mesoamerica. The famous slave markets in Azcapotzalco were originally located in the city of Cuauhtitlan. Under the Tlatoani Maxtla, however, Azcapotzalco conquered Cuauhtitlan, tore down its market, and planted the area over with cacti. In the early Aztec Imperial period, we can see that conquests often followed an economic pattern, such as conquering the "hot lands" of Morelos, which was able to produce goods -- particularly cotton -- not otherwise readily available in the more temperate Valley of Mexico. The Codex Mendoza reports that, by the time the Spanish arrived, the Cuauhnahuac area was providing the Aztecs with thousands of mantas and other cotton goods annually, as well as bulk shipments of beans, chia, amaranth, and maize.

The demand for basic goods as tribute from conquered areas near to Tenochtitlan and the other Aztec core cities fits into a pattern of using near provinces to subsidize the basic needs of those cities, particularly Tenochtitlan. As noted above, the 200,000 or so residents of that metropolis were not self-sufficient and relied upon outside supply of basic foodstuffs (a fact that would make the siege of the city so brutal). Tribute demands thus worked to subsidize the basic needs of the city. Likewise, there were tribute demands for more high value goods from further out provinces, such as gourds of gold dust, strings of greenstone, bundles of feathers, and bags of chochineal demands from Coayxtlahuacan in present present day Oaxaca.

Such tribute demands worked hand in hand with concurrent trade. Both the subsidy of basic foodstuffs and clothing, as well as of more elite goods in both raw and worked forms (and the elites willing and able to purchase such goods), led not only to general population growth, but centralization of artisans in the core Aztec cities, particularly Tenochtitlan. Not only were tribute demands making far away products like rubber readily available in the Valley of Mexico, but trade followed tribute, or really tribute followed trade.

The expansion of Aztec hegemony in an area would provide more security for Aztec merchants operating either locally or as long-distance traders, but often such an event was preceded by economic interest in the area. Aztec merchants almost invariably acted as intelligence agents, whether directly or indirectly, in an area, and as such could be greeted with suspicion and even violence, even if their business concerns were legitimate. Abusing an Aztec merchant, however, was a double edged macuahuitl. As noted above, killing an Aztec trader was tantamount to declaring war, even blocking a road could be considered an act of aggression. So areas often faced the dilemma of enjoying the economic boon of being linked into the wider Aztec trade network, but also realizing that being too rich might mean becoming a target for Aztec military conquest looking to secure and monopolize goods.