Learn Russian: Мы можем заказать какой-нибудь новый торт.

The suffix -то is used when a specific thing exists, but the speaker just doesn't know exactly which one (like 'a certain cake'). The suffix -нибудь is used when the thing hasn't been chosen yet or it doesn't matter which one it is ('any new cake'). Since we are suggesting a future action and haven't picked the cake yet, какой-нибудь is the perfect choice.

Why do какой-нибудь, новый, and торт look like they are in the nominative (dictionary) case?

They are actually in the accusative case because they are the direct object of the verb заказать (to order). However, for masculine inanimate objects like торт (and the adjectives or pronouns describing them), the accusative case shares the exact same endings as the nominative case.

Why is the perfective verb заказать used here instead of the imperfective заказывать?

We use the perfective aspect here because the focus is on a single, completed action with a result (successfully ordering one cake). If we were talking about a repeated habit, like 'we usually order cakes on Fridays', we would use the imperfective заказывать.

The English translation says 'some new cake'. Doesn't -нибудь usually translate to 'any'?

Yes, English speakers often translate words with -нибудь using 'any'. However, in casual suggestions like this one, 'some' often sounds more natural in English ('Let's order some new cake'). In Russian, the core meaning remains the same: a new cake, but we haven't decided exactly which one yet.