Learn Turkish: Kaç öğrenci ders dinliyor? - How many students are listening to the lesson?

The root of the verb is dinle- (to listen). When adding the present continuous suffix -yor, roots ending in the wide vowels -a or -e replace them with a narrow vowel (ı, i, u, or ü). Because the vowel before the ending is an i, the -e changes to -i to match, making it dinliyor.

The English translation says students, but öğrenci doesn't have the plural suffix -ler. Why?

In Turkish, whenever you use a number or a quantity word like kaç (how many), the noun that follows must remain singular. You always say kaç öğrenci, never kaç öğrenciler.

Since how many students implies multiple people, why doesn't the verb use the plural suffix (dinliyorlar)?

Because the subject phrase kaç öğrenci is grammatically singular (since it doesn't have a plural suffix), the verb must also remain in the singular form (dinliyor).

Why doesn't the word for lesson have a suffix? Shouldn't it be dersi?