Learn Russian: Мама попросила, чтобы я не уронил торт.

In Russian, the conjunction чтобы (so that / in order to) has a strict rule: if the subject changes, it must be followed by a past tense verb. Even though the potential drop is in the future, the grammar of чтобы requires the past tense уронил.

Why does this sentence use the Perfective aspect (уронил) instead of the Imperfective (ронял)? I thought negative commands usually take the Imperfective!

You are right that deliberate negative commands (like 'Don't do that!') take the Imperfective. But just like with the phrase смотри, не (watch out so you don't...), we use the Perfective for warnings about accidental actions. Dropping a cake is an accident you want to prevent, so we use the Perfective не уронил.

Could I use the infinitive and just say чтобы не уронить?

Not in this sentence, because the subjects are different: Mom is doing the asking, but I am the one holding the cake. You only use чтобы + infinitive when the subject remains exactly the same across the whole sentence, such as 'Mom carried it carefully in order not to drop it'.

Why is the verb попросила used instead of спросила for 'asked'?

English uses 'ask' for two different things, but Russian splits them up. You use спросить when you ask a question to get information. You use попросить when you make a request or ask someone to do something, which is exactly what Mom is doing here.