"When did the worship of the Theoi start in Greece?? Some sources say 4000 BCE and others say 1000 BCE, and I'm confused..."

Religion--in one form or another--has been around for at least 200.000 years, right down to the Lower Palaeolithic. From there on it has evolved, branched, been shaped and moulded. Gods formed, were conflated with others or forgotten entirely, and new ones emerged. Religion has always been flexible, so pinpointing when the ancient Hellenes started worshipping the ancient Hellenic Gods is fraught with squishy guesswork. For one, what is 'the Hellenic pantheon'? Which Gods need to have been worshipped to form the whole of the pantheon? In what form? And where? Athena was most likely imported from Lybia long before Athens was a thing, for example, so does that time count? Some of the ancient Hellenic Gods would have been worshipped in what we now call Greece, after all. The modern Greek area was already inhabited in the Palaeolithic, so does that count? How about the Minoan culture of Krete?

I think I know what you are trying to ask, though. Personally, I put the start of the worship of the Theoi at the start of the Hellenic era. The reign of the Hellenes lasted for roughly 650 years. During that time, several major changes took place within the culture and religion of these people. Trying to reconstruct all these practices is not only impractical but also impossible. As a Hellenic Recon, it therefor becomes important to find out which classical, Hellenic, period speaks to us.

Within modern Hellenic Recon, three periods in the history of ancient Hellas stand out and I'll take on the basics of each and try to explain their differences on practice:
  • Archaic Period (800 BC - 480 BC)
  • Classical Period (480 BC - 323 BC)
  • Hellenistic Period (323 BC - 146 BC)
The Archaic Period
Before the Archaic period, there was no Hellas. As the Mycenaean civilization fell, it signaled the end of the Dark Ages. The founders of ancient Hellas founded their own script, based off of the Phoenician alphabet and small social hubs began to emerge. Because the land they lived on was divided into islands, or intercut with mountains, many of these hubs were self-governed. Many wars were fought over the next 300 years or so, as the cities Athens, Sparta, Corinth, and Thebes tried to expand their land, work force and supply of raw materials.

As the population grew, the territory was expanded and colonies were set up as far as Magna Graecia (Southern Italy and Sicily) and Asia Minor. This expansion reigned in an age of tentative stability and economic prosperity. There was a lot of trade between the core cities and the settlements on the edges of the domain. It was during this time, democracy was created to arrange the rule of the city of Athens.

For Hellenic religion, this age was a formative age. Gilbert Murray in his 'Five Stages of Greek Religion' describes how the various tribes of the Dark Ages brought their Gods with them as they traveled the land and settled in different places. Various Gods with overlapping domains were worshipped in different parts of the region, forming a cohesive but unstructured whole. There are varying incarnations of Gods and Goddesses and their abilities and strength vary greatly across the land.

The Classical Period
The Classical period is the best know period. Most of what we know about ancient Hellas is from this period. The Classical period was the foundation of modern Western politics, architecture, scientific thought, literature, and philosophy. It was also the age of Athens; most of what we still know about ancient Hellas comes from records from this city who was at its greatest during the two centuries of the Classical period.

Many wars were also fought during this age, the most famous being the Persian war. Although Athens struggled through the wars and a temporary rule under Sparta, many issues settled and solidified during these two centuries. Democracy became well-defined and the major temples were built. This is the age of Herodotos, Euripides, Socrates and Plato. This is also the age in which Alexander the Great came to power.

This was also the Age of the Olympians. Many of the old Gods got merged into single personas with different epithets to accommodate local worship. This more unified faith was introduced to many of the city states and although it was never a unified whole, this was the closest the ancient Hellenic religion ever got to being a solidified faith.

The Hellenistic Period
At the start of the Hellenic period, ancient Hellas was at its largest. Alexander the Great had conquered lands as far as Asia Minor, Assyria, the Levant, Egypt, Mesopotamia, Media, Persia, and parts of modern Afghanistan, Pakistan, and the steppes of central Asia. The term 'Hellenistic', as applied to this time period is a modern invention, dating back to the mid-19th century. It is defined by the 'Hellenisation' of the conquered lands, something that succeeded only partially, although all areas fell under the Hellenic nation. It is offset by 'Hellenic', which describes Greek culture in its native form.

After Alexander the Great died, there was no logical successor. He left his empire to 'the strongest' and thus his generals fought a forty year battle which resulted in four major domains. Next to those four, much of mainland Hellas and the Hellenic islands remained at least nominally independent, although often dominated by Macedon. the four domains, called dynasties, were:
  • The Antigonid dynasty in Macedon and central Hellas;
  • The Ptolemaic dynasty in Egypt based at Alexandria;
  • The Seleucid dynasty in Syria and Mesopotamia based at Antioch;
  • The Attalid dynasty in Anatolia based at Pergam
The Hellenistic period ended with the rule of Rome over many former Hellenic territories. Both the lands and Gods were integrated into the Roman dynasty. It signalled the end of the Hellenic world as it had stood for centuries.

Perhaps needless to say, I adhere to the '1000 BCE'-theory (sort of), but really, a case can be made for pretty much any number you want.