I studied the words first (https://elon.io/learn-duolingo-english-to-turkish/lesson/quantifiers). That made it a lot easier for me.
(Onların) Herhangi biri mi yoksa (onların) hiçbiri mi?
"Herhangi bir" → "any of",
"Hiçbir →none of",
The genitive case structure: "X-in Y-i", or for compound names "X Y-i"
"Onların herhangi biri" → "any of them",
"Onların hiçbiri" →"none of them".
You are right to make mistakes at this stage but keep it up. Practice makes perfect. :)
Herhangi bir = a(ny) random (one).
Herhangi bir filmi izleyebiliriz. - We can watch any random movie. (Whatever movie you choose is fine with me.)
herhangi birİ = any random one OF THEM. (3rd person possessive)
Hiçbir = no; none
Hiçbir filmi izlemek istemiyorum. - I want to watch none of the movies.
hiçbirİ = none OF THEM. (3rd person possessive).
"Veya" is used when you're offering two choices, but other possibilities exist that you're not mentioning. Ex. Do you want to watch Batman veya Spiderman? -- You could answer something completely different, or just say "neither" or "both".
"Yoksa" is used when there are only two options... and that's it. So in this sentence, either we're interested in what's being offered or we're not... there's no other choice but these two. :-)
If you scroll up, Ektoraskan explains "herhangi birİ" and "hiçbirİ" ("of them" comes from "biri" in those phrases).
And the double "mi" is explained in the Tips and Notes: Questions 2 (scroll down to "Yoksa or Veya")
I hope that helps, but if you're still stuck I'll try to expand /clarify :-)
Why is anyone or no one wrong? I feel like this lesson is poorly written. There are a lot of possible answers that seem correct that are not being taken, and the lesson is so short and has so little explanation it's like trying to learn by trial or error. I can memorize, but not understand.
It's not easy nor well explained, that's for sure. It would help if we knew a bit more about what you were thinking. Otherwise, this sentence literally is:
Herhangi - every+which but when put together it means anywhich or any
biri - one of them (accusative because it is a specific one, and yes, I know that seems backwards because it is any specific one. Also, third person so "of them")
mi - (question word with no English direct translation but it shows that the thing it is following is uncertain or being questioned in a yes/no fashion)
yoksa - or (showing a comparison between two and only two possibilities as opposed to veya which allows for other, not listed, possibilities)
hiçbiri - none+of them
mi - (second question word with the same function as before)