The movie End of Evangelion was in no way a response to hate mail. In fact it follows rather closely earlier versions of the scripting (and draft storyboards) of the intended ending which they weren't able to complete for TV, and was made as soon as they could afford to (the preceding compilation movie Death, unsatisfactory though it is, raised enough money to finish the job). In spite of what some people say, the meaning of the ending is the same in both cases (Shinji rejects Instrumentality). There is a screen in the movie that is full of fan mail; only two of the things on it could in any way be considered hate mail, and they are both very generic (one was in fact graffiti on a wall). The English-dubbed DVDs of End of Evangelion are technically badly flawed, with seriously damaged video, such that the company that made them - Manga - is referred to in this context as "Mangle"; they are also out of print, and no English-speaking company has a current licence for the film at all. To see the film as it should look, you have to get the Japanese version, or the Italian dub, or this year's new French release (I got the Italian version and added the English dub to the file for my own viewing copy).
On the show itself - Shinji and Asuka both have serious psychological issues from their upbringing, which they fail to deal with in very different ways; Misato is pretty screwed-up, too. Rei is an artificial being, so all bets are off with her; Kaworu is another artificial being created by a different group. And what you see as mecha are actually organic beings, in spite of being controlled (usually) by "pilots".
It's worth remarking also that there are three distinct continuities - (1) the show plus the EoE movie; (2) the manga (written by the show's character designer), which was completed just this week after eighteen years; and (3) the new theatrical movies (usually referred to in English as Rebuild of Evangelion) of which the third of four was released recently. There are differences in the characterisation in the three versions, and many details are incompatible between them, so care needs to be taken in discussion.