Right now I'm wondering if きれい (kirei, clean) is related to きたない (kitanai, dirty) because I just learned that "nai" can be a negation like "orenji wo tabenai" meaning I don't eat an orange.
And in general, is there a dictionary or tool to list off inflections, conjugations, and stems of words? In this case, wiktionary wasn't useful:
In this case you can be quite sure it is not the case, because kirei is a Sino-Japanese word (it can also be written 綺麗), as indicated by your second link. Chinese morphemes in Japanese do not go through any change. The negation of きれい is きれいではない.
According to this blog, the etymology of the word きたない is きた meaning order, and きたなき (なき is a classical Japanese equivalent to ない) being the negation of that ("there is no order"). I'm not sure how reliable this source is, though, so feel free to take that with a grain of salt.
