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Screenplays and scripts primarily separate out dialogue and stage directions, etc.
Would it be feasible to write a script and mix elements of a screenplay into it, to give both technical and acting personnel a full runthrough of the whole play, alongside potentially preventing any confusion (as in the common question of "what line are we on?").
You're using terms with ambiguous meanings, which can lead to confusion. Generally a screenplay is produced by the original writer(s) of a film, and contains not only dialog, but character and scene descriptions --not necessarily in great detail, but enough that a competent production team could cast appropriate actors, design costumes and sets, and block out scenes, in accordance with the original vision (if so desired).
Script is sometimes used interchangeably with this, but I think what you're actually picturing is one of several different "scripts" that are derived from the original screenplay. The shooting scripts have elaborate camera directions added for the camerapeople, whereas the actor's scripts may be stripped down to just dialog and minimal actions --because that's what they need to deliver their portion of the film. As the original writer, you don't create these, those are crafted by the production team as working documents.
A script can also be the document used to put on a stage play. Again, in this case, the main script typically WOULD include stage directions, character and scene descriptions and so forth (individual actor's scripts might not). In any such kind of writing, whether for screen or stage, you want to put just enough descriptive material in so that it can be staged --you don't want to get bogged down in prose the final audience will never hear.
