Learn Russian: Я работал в Африке во время стажировки. - I worked in Africa during the internship.

This is the "General Factual" use of the imperfective aspect. The speaker is simply stating the fact that the event happened—they have the experience of working in Africa. They aren't focusing on the completion of the work or a specific result, just that the action took place in the past.

Could I use a perfective verb like поработал in this sentence instead?

Yes, but it changes the nuance. поработал means you "worked for a little while" (focusing on a limited duration). To simply state the biographical fact "I worked there," the imperfective работал is the most natural and standard choice.

Why does стажировка change to стажировки in this sentence?

The compound preposition во время (during) always requires the following noun to be in the genitive case. Since стажировка (internship) is a feminine noun ending in -а, its genitive form is стажировки.

Why is it во время instead of just в время, and is it related to the word вовремя?