Öğretmen, bir yardımcıya öğrencilerin ödevlerini kontrol ettirdi.

Kontrol etmek is a common Turkish verb made from a noun (kontrol) + the light verb etmek (to do): kontrol etmek = to check.
In your sentence, it’s in the causative form: kontrol ettirmek = to have (someone) check / to make (someone) check.
So kontrol ettirdi = (s/he) had (someone) check (simple past, 3rd person singular).

Not the teacher. The teacher is the one who caused/arranged the action.
The assistant (bir yardımcıya) is the one who actually checked the homework.

Because in Turkish causatives, the person who is made to do the action (the “causee”) is often marked with the dative (-a/-e), especially when the original verb is transitive.
Here the base action is transitive (kontrol etmek checks something), so the doer you delegate to is typically:

• yardımcıya (DAT) = to/for the assistant (the person instructed to do it)

If you used accusative (yardımcıyı), it would suggest the assistant is the direct object being affected, which doesn’t fit this meaning.

• kontrol etti = the teacher checked it personally.
• kontrol ettirdi = the teacher had someone else check it / made someone check it (delegated or ordered it).

Sure:

• öğrenci-ler-in = students-GEN = of the students / the students’
• ödev-ler-i-ni = homework-PL-POSS-ACC

So the whole phrase is a genitive–possessive noun phrase:

• öğrencilerin ödevlerini = the students’ homework (as a definite direct object)

That’s the normal Turkish “X’s Y” structure (genitive–possessive agreement):

• X-in Y-si
• öğrencilerin ödevleri = the students’ homework

Turkish typically marks possession twice: 1) the possessor gets genitive (öğrencilerin)
2) the possessed noun gets a possessive suffix (ödevleri)

Because it’s the specific, definite thing being checked. In Turkish, definite direct objects are usually marked with -(y)I (accusative).
Here, because ödevleri already ends in a vowel (the possessive ending), accusative appears as -ni with a buffer n:

• ödevleri + ni → ödevlerini

The form ödevlerini can be morphologically ambiguous in isolation, but in this sentence it’s effectively clarified by öğrencilerin (students’).
With öğrencilerin, the natural reading is:

• the students’ homework

bir yardımcıya means to an assistant (indefinite: some assistant). It does not specify whose.
If you want to say to his/her assistant, you’d typically use:

• yardımcısına = to his/her assistant

Example: Öğretmen, yardımcısına öğrencilerin ödevlerini kontrol ettirdi.

No. It’s optional and stylistic. Turkish sometimes uses a comma after a topicalized subject for readability, but you can write it without the comma too:

• Öğretmen bir yardımcıya öğrencilerin ödevlerini kontrol ettirdi.

Because et- becomes ett- when suffixes are added in this construction (a common spelling/pronunciation pattern with etmek compounds):

• kontrol et- + -tir (causative) → kontrol ettir-
• past tense -di gives kontrol ettirdi