Learn Russian: Мы отменили заявку за полчаса до начала. - We canceled the application half an hour before the start.

This sentence uses the structure being taught in this lesson: за + Accusative (amount of time) + до + Genitive (event). Here, за полчаса tells us the amount of time (half an hour) and до начала tells us the event (before the start). Together, they mean 'half an hour before the start'.

Is начала the feminine past tense verb meaning 'she started'?

No, although it looks exactly the same! In this sentence, начала is the genitive singular form of the neuter noun начало (the start or beginning). It must be in the genitive case because it follows the preposition до.

What case is полчаса in, and why didn't its ending change?

It is in the accusative case because it follows the preposition за in this time structure. The word полчаса (half an hour) is an inanimate noun whose accusative form is identical to its dictionary form, which is why it looks unchanged here.

Why do we use the perfective verb отменили instead of the imperfective отменяли?

We use the perfective aspect because canceling the application was a single, completed action that achieved a final result. If you used the imperfective отменяли, it would imply you were in the middle of the process of canceling it, or that you used to cancel applications repeatedly.