One is singular ("sen" form), the other is plural or formal ("siz" form).
No; it's a command, and we don't usually say the subject in commands in English. So it's just "Eat together!"
I think what is confusing me is the yiyin, if it is a command. I would have thought the sentence would be beraber ye. Why is it yiyin over ye?
You use yiyin when you are speaking to multiple people (or to one person formally). ye is for one person (informally).
If you are saying "eat together!", you're almost certainly talking to multiple people (one person can't really eat together with himself), so you need the plural form yiyin.
In what section was plural commands taught? How is a plural command constructed? Adding -IyIn to the root of the verb?
Shouldn't it be "yeyin" instead of "yiyin"? At least this is what online-conjugators like Verbix return.
Yes, but "ber" is not a too common everyday word. Instead we'd use "seene".
Iranian use Arabic alphabet, so I used the closest pronounciation in English.
Nope, that would be "Berbaer yiyelim" :) It is a different suffix on your verb.
That would be 'birbiri'. But it takes different suffixes according to the person.
Dün caddede birbirimizle karşılaştık: We met each other on the street yesterday. (birbiri-miz[since the person is 'biz'])
Birbirlerinin yüzlerine baktılar: They looked at each other's faces. (birbir-leri[since the person is 'onlar'])
Siz birbirinizi seviyorsunuz: You love each other. (birbiri-niz[since the person is 'siz'])
Interestingly, 'Birbirinizi yiyin.' means something like 'Argue(in a tiring way) with each other.', 'Wrangle with each other.' figuratively.
I thought the imperative of yemek was yer. I know it is 3rd person sing. simple present tense form but isn't it also the imperative. Is that mistaken?
Yes, indeed. Here is a table for imperative and present simple tense respectively of the verb 'yemek' :
Imperative moods of third persons are similar to 'He/She/It/They shall'.
In Turkish, imperative and present tense are different. By the way, there is no imperative form of first persons.