What is the egg mixture used in gratins called in English?
I would call it a custard, I think, if not only for want of a better word. A custard consists of milk, egg and a thickening agent i.e. flour, so it fits the description well enough.

It might sound a bit odd to English-speaking ears though. I think 'savoury custard' is probably more appropriate. I must admit I always thought moussaka et al was topped with a white/bechamel sauce; I'm pretty sure that's the more common topping in 'the West'.

As for the dishes themselves, I'm not sure if there is a catch-all English term for them - 'pasta bake' is an English version of the Italian pasta al forno, but that obviously only applies to dishes that use pasta.