What is that "would" doing in the translation? Is the sentence expressing that if you did not do your homework, the teacher would show us?
No, it's just aorist can be translated as "present simple", "will" or "would" in English.
Since there is no context here, all these options are accepted.
Why is the possessive case ödevlerini used rather than just ödevleri? Is it because the homework is necessarily yours (which the English construction might be less likely to acknowledge)?
Yes, although I admit "ödevleri" would sound better. I'll get it fixex once I'm on a pc.