Damn interesting question! Just a note, amphitheater, which means "double theater" is not the correct term to describe these structures, since it only refers to the big circular arenas and theaters that the Romans built (get it, they're a double theater because it's two theaters stuck to each other. Clever Romans and their naming...). Anyway, this is closely tied in with the evolution of Greek theater over time, so I'll have to explain a little bit about that.

Aristotle tried to discern the origins of theater, and much of what we know is based on what he wrote. Occasionally he has no idea what he's talking about, but for the most part the Poetics are good enough guesses and explanations for us to piece together what the hell happened. Greek theater derived from the dithyrambs, hymns sung to Dionysus as a ritual part of his rites. They were accompanied by various musical instruments and the singers and other participants danced around the altar. It's somewhat debated what exactly this originally might have looked like and since it's clearly the ultimate origin of various forms of theater it's often been theorized what exactly happened during the earliest dithyrambic performances (for example, the term tragoedia, which literally means "goat song," but is of course the word for tragedy, is very troubling. Were they performers originally dressed as satyrs, like in the satyr plays? Did they sacrifice a goat, and the dithyramb served in part as a sacrificial song to send the goat off and offer it to the god? We don't really know). At some point, says Aristotle, the dithyramb branched off and two separate forms developed simultaneously. One was the dithyramb, and the other was tragedy, which originated with the introduction of a single actor, equipped with various masks to sing the parts of individual characters, was added to the chorus. The Greeks generally agreed with Aristotle that this was the innovation of Thespis, who arrived in Athens late in the 6th Century and introduced the actor and tragedy to the Athenians. Aeschylus later introduced a second actor, allowing for real performances, and Sophocles introduced a third. In tragedy there were never more than three actors speaking on stage at a time, but not so in comedy--

I digress. The point is that the shape and function of the theater derives also from the dithyramb. Since dithyrambs were sung and danced in a circle around the altar, there seems to have generally been a cleared circular space in which they would be performed--the orchestra. The orchestra (which, interestingly, means "threshing-floor," which links us in with the agricultural nature of the ritual) and the altar were the central elements of the physical theater, and were really all you needed to perform most early tragedies. Sometime during the early 5th Century or possibly during the very late 6th the scene was introduced, which was that big building you see in the back of most theater ruins. The scene was originally some sort of wooden structure, or even just a tent or a backdrop that could be used to screen the actors or provide a background scene (the actors were rarely behind the scene when offstage except for much later in the 5th Century. Normally they just stood on the wings). Originally, of course, there were no seats, which is why theaters tended to be built on hillsides. At first the structure was basically an altar with the orchestra and some sort of backdrop, up on the side of a hill where people could sit in the grass and still see. We find a similar approach at most stadia, including the stadium at Olympia, which did not get seats until rather late. Seats were introduced at many theaters rather early on, probably first in wood and then very soon in marble. The Theater of Dionysus as it exists now is it's Roman form, after the Romans added a large, arched scene and a whole bunch of seats way in the back, almost doubling the size of the theater, although greatly reducing the size of the orchestra. The previous layers are still preserved to varying degrees of detail. The earliest theater seems to have been, as we expected, basically a cleared space around 25 or 30 meters wide. That's it. The Attic scene is traditionally believed to have been the first, and the first set in stone, but the rest of the theater was not carved from stone until the early 4th Century, although by the end of the 5th Century the basic layout of the Lycurgan theater more or less existed already, and it seems that all Lycurgus really did was carve the old wooden structures out of marble. During the Hellenistic Period and under the Romans an awful lot of additional ornamentation was added, such as a series of marble thrones in the front row and a wall around the orchestra, both of which probably would've been horrifying for a 5th Century Athenian, for whom the Dionysiac festivals were a ritual where rank was subordinated and in which the audience was expected to directly participate with their very presence.

So, most of the stone theaters we have today date from the 4th Century or later, by which time both tragedy and comedy had spread throughout the Greek world. Early on, however, and during the 5th Century, theaters were something of a ramshackle affair, temporarily thrown together around the altar of Dionysus. It's important to note that the theaters, or at least the altar and orchestra that made up the important bit, long predate the concept of tragedy and drama. Every city had some sort of Dionysiac ritual or one closely related that would require the singing of dithyrambs or at least paeans, and the theaters would be thrown together then. Since the altars usually didn't move they tended to be in the same place, which meant that eventually additional structures would've been put up. So, yes, pretty much every city had some sort of orchestra in which to hold the performance rituals of Dionysus and other gods, although whether actual plays were being held there is something that depends on time and place.

Well, to answer this first you need to distinguish between theaters and monumental theaters, which may be an obvious distinction but is still pretty important. The stereotypical Greek theater, made of marble and the like, is really not typical of the Classical period, defined roughly as ending with Alexander. The stone theaters at Athens and Epidauros were built in the early fourth century BCE, and a few other major sites and cities might have them, but they were really quite exceptional. The vast majority of poleis simply could not muster the resources for monumental construction on that scale. There is also a cultural aspect, as monumental stone theaters are associated with a particularly Athenocentric conception of Greek identity that didn't really begin to coalesce until the very end of the Classical period. In fact, at least in Asia Minor (I am less familiar with Greece itself) monumental stone theaters are really a product of the Roman-era Second Sophistic, and many people connect their construction with the Antonine cultural program of the second century. So if you wandered around Greece during the period we usually think of as "Ancient Greece", you wouldn't see very many at all, and Euripides, Aeschylus, Sophocles and Aristophanes never performed in them.

This isn't to say there were no theaters, but they were wooden, or in many places just a convenient hillside. There is also a question of "street theater", which certainly existed in Republican Rome (temple steps often served as a makeshift theater), as well as the "phallikos", or phallic processions that Aristotle posited was the origin of comedy (along with iambic lyric poetry). You get to sort of a tricky point here in asking how the "average" Greek experienced theater, because when we say "Greek" we really mean "Athenian", and numerically speaking the average Greek of course never saw Athens. It is sort of the weird nature of the beast that the experience we consider "Greek" really only became typical well after it was gone. Now, there is always going to be the debate about how widespread what we think of as classical culture was, and there are very persuasive arguments both for and against them being truly widespread during this period. Regardless of whether or not the average Greek ever heard Euripides, he certainly would not have done so in a stone theater.

The somewhat terrifyingly comprehensive source for this is Frank Sear's Roman Theaters. It is also worth reading Aristotle's Poetics if you are curious about this, or really just in general.