Although I’m passing familiar with Evangelion I hadn’t heard of “Death and Rebirth”, so I looked it up.
Good heavens! Summaries of the episodes from the television series followed by the first half of the movie? (Only the first half? WTF?) No wonder you’re confused. It’s like trying to understand the X-Files mytharc by watching two minutes from every other episode.
Sadly, though, I don’t think it would make much more sense even after I explained it. But let me try anyway:
First, some creature blows up Antarctica, causing massive ecological damage. This creature is called, for some reason, an “Angel”, and the destruction of Antarctica is called “Second Impact”. It is never explained in the series what the first impact was.
After Second Impact, two semi-secret organizations called SEELE (German for “soul”) and NERV (German for “nerve”) are formed ostensibly to research ways to defend against future such incidents. NERV begins designing Evangelions, which are giant artifical lifeforms that act like giant robots and are based on the corpse of the first Angel, also known as Lilith. Evangelions, it is later learned, can only be piloted by children conceived at or near the time of Second Impact. Judging from the series, it apparently also helps if the children have severe emotional damage.
SEELE spends the same period of time being very secretive, I think.
Fourteen years later, more Angels start attacking Earth with the apparent goal of reaching the corpse of Lilith, secretly in storage under NERV headquarters, and thereby initiating Third Impact, whatever that is. For some reason, they only attack one at a time, much like Borg cubes. Various 14-year-old emotional cripples are called in to pilot Evangelions to stop the Angels. Nothing but an Evangelion can actually stop the Angels, because the Evangelions (like the Angels) have something called an “AT Field” which is an invincible forcefield which may or may not have something to do with souls.
As one might expect in a military operation involving incredibly powerful weapons based on alien lifeforms and piloted by 14-year-olds, people get hurt a lot.
BUT! Secretely SEELE wants to initiate Third Impact as well, just on their own terms. This has to do with something called the Human Instrumentality Project, which makes no sense.
BUT! Secretly Gendo Ikari (the head of NERV) also wants to initiate Third Impact on his own terms, which involves Rei who is in fact a clone of his dead wife, and yes that’s as creepy as it sounds.
This goes on for about 24 episodes and then the show runs out of money and has a really stupid final episode in which it turns out to just be all Shinji Ikari’s dream, or something. Then comes “Evangelion: the motion picture” in which they give the series a proper ending which still makes no sense: Shinji Ikari initiates Third Impact on his own terms, which involves liquefying the entire population of Earth except for him and one of the female pilots.
The excessive use of Old Testament names and symbols is never explained and is apparently window dressing.
Find some good Neon Genesis fanfiction and read that instead, quite frankly; it will make more sense and will leave fewer dangling plot-threads.