The verb promise assigns three theta roles. The external role is assigned to Poirot and the two internal ones to Maigret and the job , respectively.
Last week is an adjunct.
Emsworth is walking the dog. An external one to Emsworth and an internal one to the dog . That Poirot had left disappointed the crowd immensely. Immensely is an adjunct. In the embedded clause, which acts as a subject, the verb left assigns one theta role.