Why is "my" not optional in this sentence?
Can you tell us what was your translation? The "ideal" translation doesn't contain a "my".
It rejected "Bojíme se tamté zvláštní rodiny" but accepted "My bojíme se tamté zvláštní rodiny"
That's really weird, the incubator debugger says your first translation is correct, while what you're saying was accepted is incorrect. Must be a bug on Duolingo's backend then. :(
I'm a little confused why it's "tamhlete." I see this is genitive form... So why is it genitive here? Is it because of the verb bojime? Is there some sort rule to tell us which verbs require genitive objects?
Yes, the verb bát se requires the direct object in genitive. There is no rule, you have to learn those verbs individually.