hello, I believe I can assist in answering a few of these questions. The economy of the polis would depend on what resources they have access to, and these varied depending on which polis we are discussing. What may be abundant in the Peloponnese may not be as available in Boeotia, or Attica, and vice-a-verse. I will use Athens as an example because, as far as I have read the sources describing the Athenian economy are most abundant when compared to some of the other poleis.

Thucydides and Herodotus both make mention to the economy of Athens. Herodotus accounts the discovery of a large silver mine at Laureium in late 483 BCE. Laureium is 60km SE of Athens, and as a result of this, silver became one of the main sources of revenue for Athens. Themistocles convinced the Assembly to use this silver initially to build a large Navy, with Persia in mind, but using Aegina, a small island that had always contested Athenian maritime interests as the excuse. Essentially the silver made Athens very wealthy and susceptible to trade.

Thucydides also points out that (and I'm paraphrasing here) - Attica, suffering from the poverty of its soil soon grew too large in population that they had to send out colonies to Ionia. So from this we can see that Attica, generally speaking, had poor soil for irrigation. Political implications aside, we cannot ignore that part of the reason for Athenian imperialism was for access to greater resources. The Athenian colony of Amphipolis is direct proof of this. The Black sea is considered the "Bread Basket" of the ancient world, because of the fertile soil. As you can see from this map, Amphipolis is of strategic importance for access to the Black sea, and throughout the entirety of the Hellenistic period Athens had always placed huge emphasis on maintaining this colony. Even during the hegemony of Philip II of Macedon, the Athenians were ruse'd into making concessions with the intent of retaking the colony.

For your second question, as far as the geography was concerned of Attica, the Peloponnese, and Boeotia, wood was a scarce commodity, but again there was a lot of trade that made it readily available. It should be noted that Alcibiades, who had been ostracized from Athens due to a scandal regarding vandalism, had escaped to Sparta and explained that a main focus of the Sicilian expedition was to gain access to the forests of Southern Sicily. However, Macedonia was the key to providing lumber for Greece. There are numerous treaties that we know of that had made Macedonia one of the key sources of lumber for the southern Greek poleis. In 389BCE, several poleis on the Chalcidice made a treaty with king Amyntas of Macedonia, in which they provided pitch and fir timber(Source). Thucydides states that following the Sicilian Expedition, the majority of the Athenian navy had been destroyed. Hundreds of ships and manpower were lost in the expedition. However, he explains that the navy had been rebuilt remarkably quick due to trade with Macedonia for lumber. In fact, the rise of Macedonia's neighboring polis of Olynthus was a result of it's location, where she stole prominence of the lumber trade with southern Greece. When Philip II came to power he sieged Olynthus and completely destroyed it.

-Currency and banking: I can only really comment on Athens and Macedon, but perhaps the trend is similar for the other poleis. For Athens, the silver from Laurium was used to make coinage. So for them, they had their own currency that was accepted throughout the whole of Greece. Everybody likes silver. Though it might not answer your question too well, I like this story of Macedon under Alexander which demonstrates personal coinage. The Macedonian conquests had brought in massive sums of gold, much of which were obtained after capturing Persepolis. As Alexander continued to conquer and travel further East, he left governors in his acquired territories. His appointees, thinking Alexander would never return began to exploit their power. The appointee of Babylon was Alexander's childhood friend, Harpalus, who actually began to produce coinage with himself pictured on the coins and not Alexander. Alexander was obviously furious, purged his governors, and Harpalus had fled to Athens to attempt to raise a mercenary army with the wealth he had stolen, but he was instead arrested by the Athenians and his currency was seized.

Some good sources that might be of interest, and are free on Amazon Kindle:

Herodotus, The Histories

Thucydides, History of the Peloponnesian War

Xenophon Anabasis

I also recommend Arrian, Diodorus, Plutarch, and Quintas Curtius Rufus and their books on Alexander the Great.

And finally, the speeches of Demosthenes on the threat of macedon highlight many economic issues:

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0072%3Aspeech%3D11

Greece? Why Greece!? That aside, I suppose the standard "textbook" will be The Cambridge Economic History of the Greco-Roman World (ed Scheidel, Morris, Saller). If you are looking for more social sciency approaches, particularly urban development, Ian Morris (esp. Death-Ritual and Social Structure in Classical Antiquity) is a good bet. Unfortunately I am kind of reliant on broad surveys, but I will try to mark my sources for each section.

EDIT: A quick note is that things definitely change in the Hellenistic (roughly Alexander to Augustus). If i don't qualify the period, I am probably talking about the Classical period (a bit squishier, but about Marathon to Alexander, with extreme squishiness on the earlier date). This also means that, unfortunately, this will be Atheno-centric. That damn city does seem to dominate conversation.

What was the basis for the ancient Greek economy? What were the main productive units? Family farms, large estates, market-driven plantations? I know that barley and livestock were major products--from the excellent question about Greek food earlier today--but did they have to import food? How important was fish in their diets?

My understanding is that the “classic model” Greek agricultural economy was based on what some scholars call "agro-towns", that is, villages/towns/burghs/whatever that housed an overwhelmingly agriculturally based population, and the villa never really caught on in the East. However, there were areas where dispersed settlement (ie, single farmsteads) predominated, and there would have been people who lived part of the year in town, and part of the year in a farmstead—even just within Attica. When you step outside of Attica, it varies even more, and Boeotia, for example, seems to have mostly been hamlets and homesteads. So I am not sure how far the “classic model” can hold, as there was enormous variation both between and within regions, but it might be nice as a “base”. Source is The Archaeology of Ancient Greece (James Whitley), which was published in 2001, and given the popularity of the topic will no doubt be very out of date.

Athens, at least, had to import food, but it was unusual, and I can't say to the others. Imports would have largely been from the Black Sea region, although Sicily was also important. Don't know anything about fish. Cool question, I'll look into it.

Materials: What were the main building materials? Was wood a scarce commodity (reserved for ships, perhaps)? Were stone or marble common for buildings other than large monuments and public structures? Were iron and bronze in widespread use, or reserved for military purposes? I assume that wool and leather were their main cloth-like materials; were there others?

Building material is going to be primarily stone because, as anyone who has been to Greece can tell you, there is an awful lot of stone lying around. Marble, as you say, is primarily going to be for monumental public structures, so instead you will mostly have local stone--this is an image from Rhamnous, which was a deme of Attica (I believe the wall is Hellenistic). Fired brick is largely a Roman innovation in the architecture of Greece, and I do not believe mudbrick was heavily used. For wood, I can't really say. Much of the wood for Athens' navy came from the Chalkidike in northern Greece, so it is possible that local montane wood was still used. Or, on the other hand, it could have been totally farmed out. It is worth noting that even Olynthus on the Chalkidike seems to have used primarily stone archetecture, although I still suspect that the use of wood grows increasingly important the farther north you go.

Clothing will mostly be wool and to a much lesser extent leather, which would mostly be things like cloaks, tents, shoes, etc, were the animal fibers. Plant fiber was mostly flax (ie, linen), which formed a major component of Egypt's export economy. Cotton from India and silk fro China would be a valuable luxury good in the Hellenistic period (I'm getting this from The Oxford Handbook of Technology in the Classical World--the chapters are pretty short andcover a bewildering array of topics, so it may be a good resource).

Currency and banking: Is it roughly true that each city minted their own coins, while Athenian currency was sort of a default standard? What metals did they use? Where did the minerals come from?

Ugh, this is tricky. Greek currency was bimetallic—that us, specie was in gold and in silver. Bronze, I believe, was an innovation of the Roman period, as was an economy that can be usefully thought of as "monetized" (although the Hellenistic is definitely getting there). It is true that the Athenian obol was a sort of standard in the Classical period, but there were a great number of currencies for each polis (and, of course, later Hellenistic kingdoms all produced their own coinage). This, in part, explains the existence of “trapeza” or banks/money-lenders. Much like early modern Europe, the need to change money would would been a major impetus for the formation of a banking system. That is probably too elevated of a term for the Classical period, but in the Hellenistic you do get a financial system of sorts, which I am baselining at the use of scrip. Temples, which had both the liquid wealth and imposing defenses were deeply invested in banking, but there are plenty of references to "trapeza" that seem to more or less be ephemeral stands.

Greek banking is really not my thing, but /u/Daeres is one of the weirdos who likes numismatics.

Internal divisions of labor: How was labor divided within the family and/or polis? Were there certain jobs or professions reserved for men and women? How important was slavery? Do we have any good ideas what the slave-free demographics were?

There is an article by Saller about the household and an article by Scheidel about slavery in the Cambridge book I recommended above, which will be much better than any half-assed answer I can give now. Your library almost certainly will have it, but on the offchance it doesn't let me know and I will scrounge up a few papers on this that are online.

External divisions of labor: We know that maritime trade was important, but exactly what was being traded? Different regions must have produced or specialized in different things, or else there's not much point in trading, right?

Haha, oh man, the question of the role of comparative advantage in Greco-Roman commerce is still a live one. One school of thought essentially goes with your interpretation—commerce was driven by comparative advantage and thus was essentially respondent to market forces (this is the so-called “formalist” position). Another school, however, says that maritime commerce was never extensive enough for market forces to be a true driver, as it never reached a large enough economic base, and thus commerce was essentially driven through status networks (this is the so-called “substantivist” position). Implicitly, most scholars doing economic research, at least in the Hellenistic and Roman periods, work with the former. However, there are still scholars, particularly those who take a broader view, that argue for the latter. I can talk all day about this with Rome, but I can't with Greece, unfortunately.

Does it make sense to think of the ancient Greek world as a "Free Market"?

Here is an obnoxious answer: What do you mean by "free market"? That being said, I don't think you can see much in the way of interference in day-to-day economic activity by the state or political forces.

Olive oil has been mentioned, but perhaps not emphasized enough. Judging from the numbers of amphorae found in shipwrecks, I guess that olive oil and wine were major agricultural products and also traded.

From the number of pictures of fishermen on pottery it would also seem that fishing was an important source of food.

Since this is a large question I will answer the One Big Question at the end of your post. Does it make sense to think of the ancient Greek world as a "Free Market"?

I think market exchange would be the better way to describe the commodity exchange occurring in the ancient Greek economy based on the relationship between the city merchant and the rural farmer, or hoplite. After doing a reading of Moses Finley's The Ancient Economy and other monographs of Greek trade, I learned there is a more capitalistic relationship between the countryside landowners and the city merchants.

However, it is hard to use the term use the terms "Free Market" and "capitalism" for the Greek poleis besides classical period Athens, the period after the Persian Wars. Solely because a lot more trade was done with commodities for commodities rather than commodities for money in ancient Greece before Athens flourished economically. And since "capitalism" usually constitutes trading products for money, it is suggested that Archaic Greece was not principally capitalistic in nature.

So after reading dozens of books about the ancient Greek economy, I would say that most Archaic Greek poleis overall ran their poleis on a local subsistence economy and traded when necessary. Cities like Corinth, Athens, and Greek Island poleis traded with each other in Athens, Samos, Ionian Cities, and Naukratis in Egypt. Trading that they did via the Egyptians, Phoenicians, Athenians, Corinthians, or other merchants stationed off the Ionian Coast.