Sorry if I sounded testy. Seeing badly (i.e. inconsistently) transliterated Russian is, for me, about the same as seeing spellings like "lite", "e-z", and "kwik" in English. Of course, I am not saying that there is only one way to transliterate Russian properly. There are many, many ways. Which gives me an idea for a new list!
Transliteration does not necessarily reflect pronunciation. The standard *pronunciation* of почему would be, in fact, puh-chee-MOO (to use English spelling). Here, the "o" is pronounced as a schwa (unless you're from St. Petersburg, then it's more likely to be a fairly broad ah). The Cyrillic letter that looks like a y derives, like the English y, from the Greek upsilon majuscule (Υ) but in fact is always pronounced "oo" as in "boot" (more or less) and is, in English transliteration, rendered as "u". Hence, Пушкин = Pushkin. The a and o problem is different. The Russian Cyrillic letter "o" is always transliterated "o", though how it is pronounced depends on whether or not it is stressed, where it falls in relation to the stressed syllable, whether it comes in initial position, and whether the speaker is from Moscow or St. Petersburg (Petersburgians tend to pronounce the unstressed o as a broader a than Muscovites). But the o in почему should definitely be transliterated as "o"; the word derives from the phrase по чему, which means, literally "according to what" (the word "po" means "according to"). This phenomenon of the unstressed o being pronounced as a, which is standard for Russian, is called аканье (akanye); in northern dialects, and as you get closer to Ukraine, one finds оканье (okanye), where unstressed o is pronounced o.
I had a summer intern from the Soviet Union once who had never left this stage. Enormously intelligent, but a complete pain in the ass, expecting detailed explanation and justification of all the trivial minutiae of the job. It was a long and exhausting summer for all concerned.