There is a rule of pronunciation about this. You need to say two words as ( ) +i clearly. And if you want to say three or more words of noun like (e, ke,se,te,ne,he,me,re)+i+( )( ), then you can say kind of extending the word like Eego, Seeseki(成績：せいせき）、oneesan(お姉さん）etc.

It is the same for both, you always pronounce ei as E I. It might sound like an extended EE when there is another sound after it in the word and it can also depend on who is saying it, but if you ask any Japanese speaker to slow their speaking right down you will hear both the E and the I are clearly separate sounds.

I agree with Ryoko. The correct pronunciation is "ei" when it is write えい(as:kei,sei,tei,hei,mei,rei). For many japanese it's difficult to pronunce "ei"clear (and we speak quickly). So even some Japanese think that the pronounciation "ei" clear is strange, it's not wrong. I pronounce always unclear.(^_^)

I think there's no rule. The correct pronunciation for えいご is EIGO but we generally say EEGO and it depends on the person. The same happens in the pronunciation for おう, for example, おとうさん is pronunced OTOOSAN. I hope other Japanese answer to you, especially the professional teachers.