From the sandbox, I can delete/publish/unpublish/archive the records that I've added from the administrator, but the New button is not there! Additionally, I've specified that the row have a link to edit, but that's not there neither.

Are there certain fields that must be in the table or ACL settings somewhere? Otherwise what am I missing that the sandbox Super User cannot add a record from the front end?

Do you have implemented a front-end form ?
Because by default, when you create a item layout in front, the form is not instancied, it is a FLY that comes per default.

Remove the fly, and instance a FORM brick.