1. Yes, えい and おう are pronounced like ええ and おお. I'm not sure I understand what you mean about うう, but sometimes things are spoken so fast that long vowels sound like short ones. It's just something you have to pay attention to.
2. I guess you could say historical legacy. Basically, words like とおり, where they were おを or おほ (in this case it was とほり), the newer spelling uses お. とお (十) was とを, to give another example. In cases where it was a "au" → "ô" change, like pretty much any on-yomi, the う is used. For example, 今日 (きょう) used to be けふ, and the "epu" became "efu" (sort of, really it was "eɸu"), then the ɸ disappeared, leaving "eu", which then became "yô" (I believe with a "you" in between). There's a pretty long and interesting (I think, anyway) history of phonetic/phonological change in Japanese, and it helps explain the modern spellings.
3. Eh, probably not. Since vowel length is relative, it would be difficult to tell the length of a vowel in isolation. It's like a tempo thing. You can't know whether you're hearing a whole note, half note, quarter note, etc. if there's no meter/beat put to it. I don't know if that helps you any, though.