Hmm...this phrase intrigues me- I would have thought that the sentence would be "Ποια είναι η βασική εργασία σου;" but I guess that would be wrong, so is the above simply the structure that is used- with the genitive placed between the adjective and noun? For example, how does one say "What is your favorite animal?" Is it "Ποιο είναι το αγαπημένο σου ζώο;" (not "Ποιο είναι το αγαπημένο ζώο σου;")?
You are correct, this is the structure used all the time. I don't think the other with μου/σου/του in the end is wrong, but it is strange to hear, especially in the favourite animal sentence which is something you'd hear in an informal setting. The job sentence does not ring as awkward in structure with the pronoun in the end. And it's not just when forming questions, same goes for the affirmative.
Does the word "basic" come from "βασικός"?
Yes and no.
Apparently, "basic" comes from "base" + the ending -ic.
But "base" comes from βάσις (modern Greek βάση) and the ending -ic comes from Latin -icus, which is related to Greek -ικός.
So it seems like a parallel development rather than plucking the word βασικός fully-formed.