For some reason I cannot find the adjectival spelling of "fire." "Firey" isn't right, and "fiery" seems wrong, yet my dictionary shows nothing! Helllllllp!

well, the noun is used as a modifier, as in fire truck, fire engine, etc; but fiery

"fiery" seems wrong

Fiery seems the correct form to me. Ardently, fervently, inflammably why not?

Why? Because the i and the e trade places. Why is that?

Because the i and the e trade places. Why is that?

FIRE
FIERY

Seems the i and the r swapped places. Why? If you wanted a rational orthography, you came to the wrong language. There is only aleatory happenstance here.