Am I right in thinking this construction uses the present tense to represent ‘have’ and the imperfect tense to represent ‘had’, and that no other tense is used?
yes, the part representing "have/has" would be "tener in the present" and for "had" we use tener in the Imperfecto. Those are the two tenses used with this construction. It would be very unusual to use any other tense; maybe the future:
"Tendremos escrito todo el informe antes de las seis."
We will have all the report written by six.
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