So we all know that richardson was the president in fallout 2. Well at least he was the president of the enclave. Do we have any idea who the president was before the Great War. An even more general question wold be was the president before the War, Enclave?

One of the few old world organizations to survive the Great War, the Enclave was once the shadow government of the United States. Members of the Enclave both embraced the idea of a nuclear war and knew that the common man could not survive it. They believed that as long as the "important people" of the United States survived, they could regroup quickly and wipe out communism once and for all. Though not technically part of the Enclave, many powerful corporations benefited from the Enclave's actions and their research facilities were protected during the "firestorm" of 2077.
In 2073, as the global race to exploit the handful of remaining natural resources reached its fevered peak, the United States managed to stake a claim on the world's last known supply of crude oil, buried thousands of feet beneath the Pacific Ocean.
Poseidon Oil, a major U.S. energy concern, was contracted to build an oil rig and extract the oil. Before long, the oil ran dry and it was abandoned by Poseidon.
Come 2077, with total nuclear war rapidly arriving at America's doorstep, the President of the United States and a number of other United States government officials left their posts to take refuge in a number of secret locations around the world. Among them was the Poseidon oil rig. Here, the President himself set up a secret base from which the USA could continue to exist and wage war on China, with the eventual goal of retaking the continental United States.

The Illuminati as it is belived to be, or as it actually was?

Because the Illuminati was a buch of early scientists and philosophers who banded together 'cuz the chruch would burn them for pursuing "heretical" paths.

The more you know!

Eden isn't a president in our classical sense, because he wasn't voted into office (unlike, presumably, President Richardson.)

American Presidents don't actually have to be voted in - There is a succession plan beginning with the VP, though the heads of congress, and into the cabinet (although most of this is academic... I'm not into American History, but I don't think its gone past the VP before).

So does that mean Eden is never meant to be president but he used trickery to become one? And that he is something like an overgrown A.I like Skynet from Terminator series?

Skynet is still wandering around California I think. http://fallout.wikia.com/wiki/Skynet

Anyway, there not really any evidence to support that I think. IIRC didn't he say there was no one in the facility until he signaled the remenets to regroup there?