Is there an API where I can get the member ”name” of a member (not the id member) assigned to a checklist item?
Nope. The Get a checkitem on a checklist endpoint returns only the ID of the member, exactly as per the documentation. You must then lookup the member’s name based on that ID.
Also checked and indee none of the normal undocuments nested object tricks work om checklist-items 
… If you need a lot of checklist then get all members of board and merge in memory