Why is it "подождать" but not "ждать"?
I don't know either, but here's something I found about the imperative form (...quite baffling!):
Seeing how the whole pf/ipf verb issue gets very complicated very quickly, my approach is just to accept for now, and then try to get a feeling for what's more correct.
If подождать is followed by the genitive case, why doesn't пять change to пяти?
Why is it минут and not минутов?
Because "минуты" in genitive case is "минут" . "Минутов" doesn't exist as a word.
Note that the nominative singular base word is "минута", a feminine word, and not "минут".
This sentence seems to use accusative. If it were one minute, would it be: "подожди одну минуту?"