Firstly, I shall examine the origins of humanity as explained by NGE. When Lilith and Adam came to Earth, Lilith's Lance of Longinus was destroyed while Adam's was not. The Lance's purpose is to retard evolution and growth. Hence, Lilith was free to expand and spread her LCL onto earth. From her LCL sprang all known life forms.
In NGE, there is a clear divide between one's soul and one's body. The soul is immortal. This is the reason why the soul of Yui can live on in Eva unit 01 and how the souls of dead could be gathered into Lilith's Black Egg during 3I.
Hence it is my theory that the souls of humans were born in Lilith's Black Egg. However, humanity decided to reject this form of Instrumentality and acquire AT fields. Hence the bodies of humans sprang from the LCL and their souls were attached. In Biblical terms, they ate the Fruit of Knowledge thus they gained the knowledge to realize they could exist in separate forms. However due to this sin, they were separated from the Fruit of Life aka S2 organs. Adam had nothing to do with the creation of humanity and their subsequent bodies. Humans are the offspring of Lilith and Lilith alone. This is confirmed by the fact that Kaworu calls humans, "Lilim" which is obviously a derivative of Lilith.
The Lance of Longinus retarded the evolution of Adam, thus he was unable to release his LCL. However, the souls of the Angels were born in his white egg and there they would remain until 2I. One can say that the purpose of the Lance is not to destroy the Seeds nor to reduce their ability to create. It simply stops them from creating organic beings.
When Seele discovered Adam, they removed his Lance thus allowing him to evolve. However, they tried to restrain and control Adam. This is why imo Adam appears to have shoulder restraints attached to his body.
Adam probably interpreted Seele's actions against him as acts of aggression, thus he decided to wipe out these foolish creations of Lilith by self destructing his body and freeing his Angels Seele then used the Lance to try to stop him. With this massive amount of energy, evolution sped up thus creating the bodies of the Angels. Adam's body was reduced to a fetus and his white egg was destroyed. However, his soul was separated from his body and injected into the Angel Tabris aka Kaworu Since the Angels were still in a form of Instrumentality when they were rudely awakened, they retained the forms that they imagined themselves to be during Instrumentality and they kept their Fruit of Life.
Here's where things get a little tricky. For some of the Angels, they simply wanted to be reunited with their father Adam. For others, they may have desired to join with Lilith to create a 3I and wipe out the humans. Regardless of their intentions, Nerv decided to wage war against the descendants of Adam and then use Adam as a tool to be reunited with their mother.
Eva unit 01 was created to be the new Adam. It was removed from the body of Lilith the mother of humanity rather than Adam. Humanity wanted to create a new Adam that they could use to bring humanity to its former state.
The Seeds of Life are death to each other. Adam's power can break down the AT Fields of humans. Nerv needed this power to remove humans from their bodies, but they also needed their God Lilith to gather their souls together once more.
Thus the descendants of Lilith engaged the descendants of Adam in war. In the end Lilith was victorious but the new Adam that Nerv created chose to reject Lilith and Instrumentality. This is the irony of 3I and Seele's plans. In the end, it was the god they created in Lilith and Adam's image that would be the death of Adam, Lilith, and most of humanity.
In conclusion, can we say that Adam and Lilith were at war with each other prior to 1I? To be honest, we cannot tell for sure. The Gods seem to often fall victim to circumstances, Lilith loosing her Lance, Adam being awakened by humans. However, we may be able to gain some insight into their character by their circumstances and actions.
For one, Adam seems to be the God of Death and Judgment. He attempts to eradicate the sinful human race and his very presense causes death. During 3I, he is used to wipe out the human race. One might say he is like the God of the OT, sending out his messengers to judge the sinners.
On the other hand, Lilith is the God of Rebirth. She is the one who brings humans back into her womb. The imagery surrounding her is quite telling. She is crucified and her blood gives life. She wants to forgive the sinners and bring them back to their former state. This reminds me a lot of Jesus. Thus, one could say that Adam and Lilith are involved in a conflict due to their conflicting natures. However, Kaworu's existence may pose a threat to his theory. He has the soul of Adam within him, thus he represents Adam's character. He simply allows the Lilim to gain control of 3I once he realizes that the object down in Terminal Dogma was not Adam but Lilith. We can interpret his actions in a few ways. No 1: He simply wanted to give the Lilim a chance to perform their plans. No 2: He is simply apathetic. No 3: He did not want to be joined with Lilith because of his conflicting nature aka conflict with Lilith. If the third option is true, then the War between the two seeds is true. One can say the same thing about Rei's character. She is self sacrificing at the core and in the end she decides to give the keys of bliss to Shinji because of her love. She is also highly maternal which goes along with the character of Lilith being the mother of humanity
However, the Gods are not the main players here. It is their offspring who wage the wars and make a new God. Hence, they are the ones who should be evaluated intensenly, more so than the Gods.
Please excuse any spelling errors.(I think that's all. I'll add more to my theory if I think of anything else).
) Here's a couple of additions, where I shall sin and actually use quotations.