Do multiple Gray Merchant of Asphodels trigger each others' enters-the-battlefield effects?
No, only the one entering the battlefield triggers.

Any time you see a card use its own name in its rules text, that means it is referring to the card that it is printed on; not other cards with the same name.


  201.4. Text that refers to the object it's on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.