"The architect builds the stage."
Translation:Architectus scaenam construit.
Yes, it has to do with which person is doing the action, the noun cases change based on how a noun modifies the sentence. Architectus is nominative and is used when the 'architect' is doing the action (it is also used with forms of esse).
Architecto is ablative and is used since the preposition cum is used (cum architecto : 'with a/the architect').
This brings up a few questions:
Do all job titles agree in gender with the person who holds them, or do some job titles just have their own grammatical gender irrespective of who holds them? ("La guardia" in Italian is a good example of a job title that is grammatically feminine irrespective of who holds the position.)
Are we being taught Classical Latin as it was spoken (in which case, were any women doctors or architects back then?) or are we being taught modernized Classical Latin?