Learn Russian: Смотри, не разбей мой любимый стакан. - Watch out, don't break my favorite glass.

You are right that regular negative commands, which are prohibitions, use the imperfective aspect. However, this sentence is a warning about an accidental action, not a ban on an intentional one. When warning someone not to accidentally do something, we use не plus the perfective verb. The phrase смотри, не is a very common trigger for this rule!

Is смотри being used literally to mean 'look' here?

Not exactly. While смотреть means 'to look,' in this context смотри acts as a warning word meaning 'watch out,' 'be careful,' or 'see to it.' It sets up the warning structure for the next verb perfectly.

What would the sentence mean if I used the imperfective не разбивай instead?

If you said не разбивай мой любимый стакан, it would sound like a strict prohibition against a deliberate act, as if someone were intentionally trying to smash your glass! By using the perfective не разбей, you acknowledge it would be an accident and are just telling them to be careful.

Why does the object мой любимый стакан look like it is in the nominative case instead of the accusative?

It actually is in the accusative case! For masculine inanimate nouns like стакан (glass), the accusative form is identical to the nominative. The modifiers мой and любимый follow this same rule, which is why the whole phrase doesn't change its spelling.