Can the game be beaten on the first run, and is there any special interactions if you do so?
As was discussed in the comments, a video showing a run that beats the game quickly on the first run on a fresh file exists, with developer commentary over it mentioning some unique dialog.
Although the conversation mostly proceeds as normal, after Hades mentions that he always kept his temper, Zagreus' response is slightly different.
The usual text box, with added bold on the part that changes:
Well, let's see. You've berated me repeatedly and often. You've lied to me. Mustered your wretched forces to kill me, over and over. But, sure, I'll grant you that you've always kept most of your anger bottled up. All that's about to change, then?
While getting to Hades on the first run will cause Zagreus to have this response instead, with added bold on the changed part:
Well, let's see. You've berated me repeatedly and often. You've lied to me. Mustered a bunch of incompetent wretches to try to kill me. But, sure, I'll grant you that you've always kept most of your anger bottled up. All that's about to change, then?
I think there are three reasonable points at which this game could be considered "beaten". The first one can be described spoiler-free, the other two are heavily spoilerish.
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Finishing a run and defeating the final opponent.
This is the one that Joseph answered: yes, it can be done on a perfectly fresh run. All that's required is skill and some luck.
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Credits: bringing your mother back to the underworld.
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Epiloge: having the great reunification feast with all the gods.
The second and third option can never be finished on the initial run, since they require several conditions that can't be completed in a single run, such as talking to certain characters multiple times and talking to certain gods that don't show up in the first few runs.
A theoretical fourth "completed" condition would be fully unlocking/upgrading everything, but I don't think that has any kind of story implications and is purely a completionist goal.