Learn Russian: Студент продолжает редактировать текст, чтобы закончить задание. - The student continues to edit the text in order to finish the task.

This is the core rule of phase verbs (verbs about starting, continuing, or finishing an action). After a phase verb like продолжать (to continue), the following infinitive must always be imperfective. This is because you are focusing on the ongoing process of doing the action, rather than a finished result.

Why is закончить perfective here? I thought phase verbs went with imperfective verbs?

закончить is indeed a phase verb, but here it is the main action of the second clause, taking a noun object (задание) instead of another verb. The student's goal is to successfully complete the task as a one-time result, which requires the perfective aspect. If it had a verb after it, like 'to finish writing', that following verb would have to be imperfective.

Why do we use чтобы before закончить?

In Russian, чтобы followed by an infinitive is used to express purpose or intent, translating to 'in order to'. It connects the main action (продолжает редактировать) directly to the student's goal (закончить задание).

Why don't the endings of текст and задание change, since they are receiving the action?

Both words are actually in the accusative case because they are direct objects. However, текст is an inanimate masculine noun, and задание is a neuter noun. For both of these categories, the accusative case form is identical to the nominative (dictionary) form, so their endings don't appear to change.