1. The spelling does not really correspond to the pronunciation.
There are no diphthong allowed in ancient japanese. The Chinese loan words and the phonetic change of は行 braked this rule, then diphthongs appeared.
All the diphthongs listed above, together with ああ いい うう ええ おお, can't be regarded as two separate vowels. For example, the length of these sounds is not as long as 2 Mora. Usually you can't have an accent on the second vowel...
The suffix of verb and adjective are not regarded as a part of diphthongs. But う音便 seems to be.
3. It seems that Japanese do not really distinguish between ei/e or ou/oo
Although these sounds do really exist in their language, it seems that they don't distinguish between them.
I have checked my NHK日本語発音アクセント辞典, The pronunciation of えい sound in kanji is written as エイ. However it's actually pronounced as エー. I don't know why.
Actually, Japanese vowels are very loose.
e can very from [e](sometimes even [ɪ]) to [ɛ].
o can be [o]~[ɔ].
u can be something between [ʉ] [ɯ] [ʊ]
It's even possible that a monophthong is pronounced like a diphthong. There are questions on Japanese forums in China every month such as "Should お be read as ou, o or ao"(Chinese ao is close to [ɔ]or[ɔʊ]). We often call it 月経スレ.