"Are any of you going to the park?"
Translation:Herhangi biriniz parka gidiyor mu?
It is a type of possessive construction - pretty much a literal equivalent to "any of you" in English. Herhangi - any, biriniz - one of you. This is also why the verb is singular - because herhangi is the headword. So while spoken English often uses "are" for this expression, the most correct would be "Is any one (of you) going to the park?"
I think, in my wholly unworthy opinion, it would have to be "sizin herhanginiz" because as well as genitive case you need the possessive suffix as it is YOUR any...but I am not fluent or native and only started learning Turkish like three months ago so this is probably bad advice.
"Sizin biri" is not correct in Turkish. You can however say "sizin biriniz", "sizlerden biri" or "sizlerden birisi". Please note that "sizin biriniz" is less likely to be used, it is not very common.
"Sizlerden birisi parka gidiyor mu?" has the same meaning with the sentence.
In this sentence you must have herhangi because it is part of a construction with biriniz. The literal translation of "herhangi biriniz parka gidiyor mu?" - Is any one of you going to the park? vs. "Kimse parka gidiyor mu?" - Is anyone/someone going to the park? The first question is about one or more members of a group of people you are asking, the second is a question about no one in particular. It is harder to see the difference in English because of the similarity between the pronoun "anyone" and the construction "any one (of x)."
What it "is" what? I don't understand what you are asking.
Sizin is a genitive construction meaning it is possessive. The English sentence being translated is not genitive. If I try to work backwards your translation seems to me to be "Are your any of you going to the park."
I agree that biriniz looks like it has the possessive suffix indicating a genitive construction. But Turkish doesn't seem to think it is genitive. I got confused on this because I couldn't understand why it was mu and not musunuz. I believe this is the same thing - if sizin was correct then musunuz would be correct too, and it wasn't. The trick, if I am understanding it is that "herhengi biriniz" just isn't (or isn't treated as) a second person subject. It is treated as a third person (singular too I think) subject.
But this whole lesson is super confusing and not well explained.
If the ending -niz at biriniz is not possessiv, what case is it then? I actually think it is possessive (see for example also the comment of natemarz)...
If it is possessive it wouldn't have to be -musunuz either because the "anyone / herhangi biri" is the subject" and that 3rd person singular. So it would still make sense that it's "mu", even though I use the possessive form. For example "Is your cat eating / Is the cat of you eating?" "Sizin kediniz yiyor mu?"