DHARMA Initiative’s Goals and Presence on the Island are
To research the Island's supernatural properties
While not excluding other objectives, researching the Island's supernatural properties may have been one of the main objectives. Various DHARMA logos carry strong allusions to the Greek god Apollo (Jacob?), who has been "variously recognized as a god of light and the sun; truth and prophecy; archery; medicine and healing; and more. As the patron of Delphi (Pythian Apollo), Apollo was an oracular god — the prophetic deity of the Delphic Oracle. Medicine and healing were associated with Apollo, yet Apollo was also seen as a god who could bring ill-health and deadly plague as well as one who had the ability to cure. Amongst the God's custodial charges, Apollo became associated with dominion over colonists, and as the patron defender of herds and flocks." (excerpt from wikipedia.org)
As such "benevolent" colonists the DHARMA Initiative was tolerated for some time by the Others, in contrast to the US expedition intending to abuse the Island as a nuclear test site.
During their research the DHARMA Initiative eventually discovered that keeping the Man in Black prisoner was one of the Island's major functions (hence the warning Egyptian hieroglyphs in the Swan Station's computer room), knowledge that the Others considered to be dangerous and ultimately resulted in the Purge.
To design a program to improve the quality of life
A program designed to improve the quality of life of human beings by accelerating the path to enlightenment.
DHARMA consists of scientists with an egalitarian ideology:
The initiative is based on the notion that with scientific experimenting they can create a utopian society where everybody is given an equal opportunity in life. Equality for them means that everyone is depending on each other to the same degree and that no one is redundant. They believe in the duality of existence, that all human beings have a need to experience domination in its completion, both being the one in power and the one subjected to power. They also believe that all human beings have a desire to depend on someone and at the same time wants someone to depend on them, that these are basic human needs. In fact, so important to us are these needs that if taken away, life loses its purpose.
It's a huge stretch to consider Dharma "egalitarian", Dharma chooses what opportunities individuals have - If Dharma decides you're going to be a janitor, you're a janitor and you stay a janitor, no appeals process. There is no basis in the show for the assertion that Dharma believes people should experience "being the one in power and the one subjected to power" , there is a clear power structure and the ones on top stay on top. It's not like Horace Goodspeed swaps jobs with Roger Linus once in a while...
Assuming what is known via The Lost Experience is canon. Knowing that the Valenzetti Equation was pointing toward the end of the world, the island was chosen by the DHARMA Initiative to essentially create a "world laboratory". The different stations were established to modify different attributes on the island. While the stations proceeded in their research, the Valenzetti Equation was run every 108 minutes to see if they had managed to change the numbers. The terminal in the Swan was used to broadcast the numbers so that DI would pick it up in a listening station off-island.
The DHARMA Initiative may have been brought to the Island by Jacob. This would appear to be contradicting with the fact that Jacob is normally associated with the Hostiles/Others. However, it could be related to the ongoing rivalry between Jacob and his nemesis. We know from dialog between Jacob and his enemy that there have been people on the Island before, most likely the people who built the Temple and other ancient structures. There may have been a fight among them, as his nemesis pointed out. Because there weren't any of the original inhabitants left, Jacob decided to bring the Black Rock to the Island to start over again. As the years passed, Jacob's nemesis found a way to prevent Jacob from communicating with his people. This gave his nemesis the chance to impersonate Jacob and thus control his people. Jacob's response to this was to bring a new group of people to the Island. The DHARMA Initiative had something in common with the ancient inhabitants, but was also very modern. DHARMA maybe had more knowledge on the Temple and Monster than the Hostiles. This explains why they build the Barracks over the summoning chamber, why they were the only ones who had a name for the monster, why the Temple had a DHARMA logo, why the countdown timer had hieroglyphs, why Horace Goodspeed would build a cabin for Jacob, why Paul would have an Ankh necklace, and why the Frozen Wheel has the shape of an octagon, like the DHARMA logo.
The DHARMA Initiative may have eight stations. Each DHARMA logo has the same eight characters surrounding it, and each character is unique. Also, as revealed in the Lost Experience, the Hanso Foundation is currently working on a Spider Protocol. Spiders have eight legs, which works with the theory. Plus, if you add all the possible stations in the blast door map together, including the ? in its center, there are 8 known stations on the Island.
Possibly, there is one station for every monitor in The Pearl, thus, we have 9 stations on the Island.
Plus the Pearl itself, so that makes 10 stations.
Some of the notes on the blast door map might refer to meteorological experiments conducted by the DHARMA Initiative.
"Mountainous terrain most likely used by D.I.H.G. for meteorological research due to..." +
"Geological composition likely to cause magnetic interference with weather project"
The Door may also be scribbled out due to its destruction or damage. The Others may have simply found the door from the station and used it in the Decoy Village.
The Door could have possibly been a station but the hostiles may have been too much of a threat. It was abandoned, and built somewhere within the truce boundaries.
Since the Blast Door Map was mostly drawen by Radzinski, the scribbled out station on the top left corner may be the Orchid due to his unawareness about the function of that station. The question mark in the middle was put also drawen by Radzinski due to his inability to know its occupants.
Aside from the meteorological facility, a mention about a manufacturing facility for light industrial equeitment is also mentioned on the Blast Door Map, possibly giving a clue to the Tempest.
Back to the idea that there may be ten stations on the Island, there is the ARROW, SWAN, FLAME, PEARL, HYDRA, ORCHID, TEMPEST, STAFF, and LOOKING GLASS. That is nine so far, but what is the tenth? Possible tenth substitutes may be the "DOOR", or the "LINE". The line logo was seen on the shark's body. A third station logo would be the "TEMPLE" seen on Ben's map. For now, there are only nine cataloged stations that exist.
The Man in Black influenced the DHARMA Initiative's creation
The DHARMA Initiative is simply a convoluted and complex (but necessary) vehicle for Jacob's nemesis to create the conditions needed to destroy Jacob. Jacob's nemesis promised to find The Loophole, and after actually conceiving of a solution, he created the DI as the method to manipulate people until one of them could be influenced to want to kill Jacob (to be a premeditated murder, not simply an accidental/unintentional killing). Jacob's interactions with those involved (as seen in flashbacks) at stages prior to their involvement with the DI is Jacob's attempts to encourage the good in people and to plant the seeds that in the future may prevent Jacob's nemesis from pressuring them to kill Jacob. All theories and motivations behind the DI are simply creations by Jacob's nemesis (even if true, like the Valenzetti Equation) to facilitate the manipulation of humans for his purpose of destroying Jacob. The Others are people working at the behest of Jacob to counter his nemesis' efforts.