Part 1:

This is very hard to answer because it's so big. I'll start with The Maya, then discuss the Aztecs, then the Olmecs, then I'll go into your other specific questions. First, a note about timing. Mesoamerican chronology is divided into various periods. The first relevant period for our purposes is the Preclassic, which began around 2000 BCE and ended around 250 CE. This is the period when Mesoamerican civilizations began to acquire their hallmarks, and the first Mayan and Olmec civilizations started. The next period is the Classic, which ranged from 250 CE until 900 CE. This period is held by many as the apex of Mesoamerican culture, with the flourishing of great civilizations such as Teotihuacan and major cities of the Maya Area such as Tikal, Palenque, and Calakmul. The final era is the Postclassic, ranging from 900 CE to 1521 (guess why). This era is marked by a relative decline of the Maya Area, though large cities like Chichen Itza and Coba still flourished. Central Mexico became increasingly dominant, culminating in the Aztec Triple Alliance.

The Maya established countless city-states over millenia but were never unified by a single political entity. Their civilization can be said to have begun around 1000 BCE in the Preclassic in the Highlands of Southern Guatemala and lasted independently until 1697 when the final one (Nojpetén, on the island of Flores in Lake Peten) was conquered by the Spanish. Since Mayan civilization lasted over 2500 years it's difficult to condense specific features into a few sentences, but some key characteristics were similar monumental architecture (which of course evolved over time), use of the fully functional Maya Script (starting in the late Preclassic, around 300 BCE), use of two interlocking calendars (one ceremonial, one secular), use of the Long Count system of notation to mark time, and a great focus towards mathematics and astronomy. A form of the creation story in the Popol Vuh (known from Colonial era texts) seems to have been common across the Maya Area as well reaching as far back as the early Preclassc. They shared many features with other Mesoamerican cultural groups such as the Olmecs and the Aztecs, including the Mesoamerican Ball Game, human sacrifice, autosacrifice (an offering of one's own blood, non-fatal), a host of deities (under different names of course), similar crops (especially a heavy emphasis on corn), monumental architecture (step pyramids were common), a heavy focus on space and time (especially the four cardinal directions), and the Mesoamerican calendar I already mentioned (though it's not known that the Olmecs had this).

The "Aztecs" are actually not a thing, but presumably you were talking about the polity known as the Triple Alliance of Tenochtitlan, Tlacopan, and Texcoco, which lasted from 1428 to 1521. The primary ethnic group of this empire was Nahua (who spoke and still speak Nahuatl), though it contained dozens of them and hundreds of languages, including many Mayan languages. This empire was aggressively expansionist and hegemonic, demanding heavy tribute from conquered territories in the form of goods. They were the primary political power of their time, though they had enemies such as Tlaxcala (who were also Nahua) and the Purépecha Empire (who were not). They possessed all of the hallmarks of Mesoamerican cultures that I mentioned earlier, though they of course had their own unique cultural traits and incredibly evocative artwork such as this awesome statue of the god Xochipilli. They were known for having accelerated the tradition of human sacrifice, though almost certainly not to the absurd lengths described in records of conquistadors. They didn't possess writing, though they had a symbolic script that could be used to mark proper nouns via the rebus system, and was likely in the process of developing into a fully functional phonetic script.

Significantly less is known about the Olmecs, but a great deal more than you might expect is still known due to the efforts of dedicated archaeologists in the past century. While the Maya resided in the Maya Area and the Aztecs ruled from the Valley of Mexico (now Mexico City), the Olmecs created their own thriving civilizations in the Isthmus of Tehuantepec (Mexico's skinny little neck). The first Olmec site is considered to be San Lorenzo, which formed around 1400 BCE, while other major sites like La Venta formed later. These cities contained large ceremonial centers like later Mesoamerican cities would, as well as the beginnings of monumental architecture. La Venta contained a huge earthen mound that was likely formed into the shape of a stepped pyramid, though it wasn't covered in stone. The Olmecs performed human sacrifice, grew corn (and other Mesoamerican crops), almost certainly played the Ball Game (rubber balls have been found), and worshipped early forms of many of the gods later Mesoamerican cultures would worship. Because of these facts they are considered the progenitors of many Mesoamerican cultural features. They may have been the first to have played the Ball Game, and may have created the calendar systems (though it's not known that they used it at all). This chart by Miguel Covarrubias, one of the discoverers of the Olmec Civilization, shows how the Olmec storm god (at the bottom, often called the "were-jaguar") evolved his appearance over time throughout Mesoamerican civilizations. This god was known as Tlaloc to the Nahua and Chaac to the Maya. Like the Maya, the Olmecs didn't have a single empire, but rather their culture was spread across numerous city-states over a long period of time. There are findings of Olmec artwork across Mesoamerica, as far as the Valley of Mexico or the state of Guererro, and they extensively worked jade sourced from Guatemala, so their trade routes were wide-ranging. Olmec influence across Mesoamerica is difficult to interpret, and theories have ranged from military domination to cultural transmission through trade. The latter is considered more likely at the moment. After the decline of the early Olmec civilizations such as San Lorenzo and La Venta, the Olmecs entered the "Epi-Olmec" phase and persisted in numerous smaller sites until around 250 CE. Aspects of this later civilization would blend into early Mayan cultures, with the site Izapa in Chiapas being seen as a major one that contains features from both. Unlike the early Olmecs, the Epi-Olmecs have scattered evidence of what is unquestionably writing, and though it superficially resembles the Maya Script, it is undeciphered, and will likely remain so forever.