Learn French: C'est la critique qui a provoqué cet échec au box-office. - It is the review that caused this failure at the box office.

Yes! The gender completely changes the meaning. La critique means the review or the criticism (as a concept). If it were masculine (le critique), it would mean the critic (the person writing the review). Since it is feminine here, we know it refers to the written review itself.

How does the verb agree after the emphasizing phrase C'est... qui?

In sentences using C'est... qui for emphasis, the verb following qui must agree with the noun or pronoun right before it. Here, qui refers to la critique (third-person singular), so the verb is a provoqué. This rule is strict for all subjects, which is why you would say C'est moi qui ai provoqué (It is I who caused).

Why is the demonstrative adjective cet used before échec instead of ce?

Échec is a masculine singular noun. Normally, you use ce for masculine singular nouns, but because échec starts with a vowel, you must use cet to make it easier to pronounce and avoid a clash of vowel sounds.

Since box-office is a borrowed English word, how does it get its gender in French?