Interesting, are the words çift (pair) and çiftçi (farmer) of the same etymological root?
According to the etymological dictionary, çiftlik (farm) is a field that is ploughed with a pair of oxen. Hence the word "çift" getting used. Çiftçi is the person that works in a çiftlik, so there you go.
Why wouldn't you say, "ayakkabılar"?
you cannot use plural after "bir çift", similarly, you cannot use plural nouns after numbers, or many other things that indicate plurality (2 ayakkabı, 5 ayakkabı, birçok ayakkabı, etc)
Is the "ayakkabı" grammatically possessed by çift?
No, it has the suffix because it is a noun compound
ayak (foot) + kap (container) = ayakkabı (shoe)
Is "cift" the same as "iki kere"? They both mean pair/double.
No. iki kere means two times, twice, not double.