No, the EIR is applicable to any aeroplane for which a class or type rating is held.
Well yes, but you can't get a class or type rating on a single-pilot high performance complex aeroplane or a multi-pilot aeroplane without an IR.
Actually, you're right, it's a pre-requisite. So if your IR has lapsed but you still have the relevant type rating, I guess you could use an EIR.