Some statements you have made about the Japanese Language are ambiguous at best. Let's clear them up, but first let's be precise with the terminology. Pronunciation and sound have not interchangeable meaning. Every sound -or sequence of- in Japanese is associated with a specific character. It happens that some Kanji have multiple sounds or that a group of Kanji shares the same sounds, but still how are read it's univocal. There is no pronunciation involved. Just to give a quick example, the sound of あ is one and only one, while in English the sound of `a` has different pronunciations in pal, Paul, paediatrics, and so on. For the sake of simplicity it's ubiquitous savage practice overlapping the use of the "reading" with the one of "pronunciation" and the one of "intonation", but you should know the difference. The pronunciation transforms based on the preceding and/or following characters. There's no such concept in Japanese. The sounds of the Japanese language are distinctively dictated by Hiragana. Therefore what you have is a "Reading". A reading is made of one or more sounds. But once again readings or sounds have no multiple pronunciations. I hope I was able to shed some light on this complicated subject. It's a nuance, but makes an important impact.

Quoting: "I should note that there are two different alphabetical sorting orders in Japanese. For this article I am going to use the a i u e o (あいうえお) sort order."
Alphabetical order as we know it, it's one in Japan too. The order of the Hiragana works exactly like our alphabetical order: a preordered sequence of scripts from A to Z is the methodologically equivalent to the Hiragana from あ to ん, albeit there is no words starting with を and ん - last two characters of the alphabet. The Katakana alphabet is exactly like Hiragana, just the characters `design` changes - it is only used to write, in Japanese sounds, imported words from non-Japanese languages (Chinese and Korean are exception because you can use the Kanji to write them, although when writing just Chinese or Korean sounds, you will be using Katakana still).
When talking about `ordering` we normally refer to which sequence we are listing the Kanji: by their Hiragana sounds is the most common way of listing them. They can also be listed by what in English are called Radicals - foundational shapes composing the Kanji ideograph, they can be listed by strokes count or be listed by their Chinese reading - the Japanese sound of the equivalent Chinese Kanji, or even by recurrence.
In the Japanese language it's frequently used `ordering` by meaning, by which Kanji may or may not be listing, where only Hiragana is used to write all the words. This method is the closest rendition in Japanese to what we are accustomed to name English dictionary. More here https://en.wikipedia.org/wiki/Japanese_dictionary
I think it's already clear where the problem of sorting lays. Kanji, Hiragana and Katakana, though being different looking alphabets, have a well-defined scope and interconnection in the Japanese language. Kanji are words, thus have meaning; Hiragana offers a vocalization to the Kanji and also absolves the crucial grammatical role, providing the language with adverbs, prepositions, particles, determiner, verbs conjugation and more. Katakana usage is sidelined to just words foreign in nature, hence they exclusively represents sounds, carrying no meaning. Consequently when reading Japanese on a generic subject Kanji are mostly encountered, some Hiragana that connects them are present (it needs to be said that very common words are often written in Hiragana only, e.g. こんにちは means Hello) and sporadic Katakana when some imported word is used.
This is happening because in the Japanese language there is hardly any punctuation at all - yes, there's a full stop, a comma, a way to encapsulate direct speech, but no concept of empty space. You can read Japanese from any direction you set up yourself for. Traditionally it's read top to bottom from left to right. Modern books are read as ours would be, even so sometimes you turn the pages backward to further the reading, e.g. comics etc.
The English alphabet is used in case of very technical terms or for proper names - the context may sometime better served with romaji. But that's not a rule whatsoever. Katakana is employed more often. The Western alphabet -romaji in Japanese- adoption is similar to how you would take up French, Spanish, Italian or German words in your writing. (continue)