Quoting from "Sorting Settings":
"In this example you can see ABC and katakana are separated. Kanji are then separated from katakana. There were no hiragana in this list[...]"
The Hiragana on that list are the characters と and の. On that particular list の expresses the meaning of correlation and と convey the mere meaning of `and`. For instance 地域 と 言語 の オプション (notice I myself have entered the `spaces` for clarity's sake, but there should be none) is a very instructive example. Those are actually three distinct nouns you are trying to order as one: 地域 reads ちいき (Hiragana for `chiiki`), means area; 言語 reads げんご (Hiragana for `gengo`), means language; オプション read opushon, is, you guessed, option in English. Note I didn't write that オプション means option. オプション IS the word `option` written with Japanese sounds or, in other words, written in the Japanese script - i.e. Katakana, you guess it.
So 地域と言語のオプション can be translated as "Regional and Language options".

Quoting from "Sorting Names"
"It is very possible to have different people with the same name write their name in different character sets. The traditional way of writing the Japanese name of Ayumi would be written in kanji; a modern, stylish way would be to write it in hiragana, and a second generation Japanese-American might write their name in katakana or the alphabet."
Japanese always writes their name in Kanji. They don't use different sets. When the Kanji composing their names can assume several sounds, they write alongside what's called Furigana, the Kanji reading - usually as a subscript or superscript. Furigana is written in Hiragana (not Katakana as you stated later on), but for the Internet where websites could potentially be read by a non-Japanese crowd, Katakana might be used in some cases. Nonetheless as I said earlier Hiragana and Katakana differ only by scripting style, if you wish to call it as such. So which script is in use, it's not a relevant issue for foreigners.
All the other statements are a matter of opinion, save for the last: Japanese do write sometimes their names in the romaji, aka "ABC alphabet", especially when they are dealing with foreigners at any level. Although why, being 2nd gen Japanese-American entails writing its own name in Katakana, beats me. It's a tiny bit like saying, forgive me here, a 2nd gen Italian-American would write its name in Latin.

Quoting from "Kanji - The Real Problem":
"Kanji have multiple pronunciations, determined by the context in which it appears.[...]Only from the context in which the kanji appears do you know how to pronounce it."
That's like saying the pronunciation of the word 'pool' is determined by the context referring to water or balls. Not the pronunciation, but the meaning of the word is changed by the context. We all agree on this statement.
Single Kanji words change along with their meaning based on the context, but like the word 'pool', its reading is the same. e.g. あめ (reads ame) which means 'candy' if written 飴 or 'rain' if written 雨 . In this case the Kanji itself controls its meaning and reading, not the context. When single Kanji is a verb the meaning and reading changes with what's called Okurigana - Hiragana written after the Kanji absolving the purpose of conjugation -, while the Kanji remains invariant. For example 着く and 着る reads つく (tsuku) and きる (kiru) respectively, the former means 'to arrive' and the latter 'to wear'. Single Kanji aren't at all like you describe them.
Compound Kanji words follow a different ruleset build upon how many they are. Generally speaking one kanji in two-Kanji words has multiple readings depending on what is the word it appears in and where it appears in that word. You can learn the rules, or you can get used to them just by seeing them used in massive frequency. This case alone is as you've described: the need to know the context in which the Kanji lives. But since the meaning also changes with its reading, you could be able to catch the overall meaning of the sentence without being able to read that single word. But compounds Kanji with multiple readings aren't that recurring and they generally represents common words requiring not much effort to memorize.
Compound Kanji words composed by more than two Kanji reads unequivocally in one way as English words do with very few exceptions.
More here https://ja.wikipedia.org/wiki/%E5%90%8C%E5%BD%A2%E7%95%B0%E9...
If anything what could, quote "[...]keeps students up nights studying for years[...]" isn't the multiple Kanji readings, but the fact that you need to know between 2000 and 3000 Kanji and its their combinations that build words (mostly two Kanji words). So it's like having a permutation with repetition (in ordered arrangements) of 3000 syllables that makes words in pairs or singularly.

Quoting "Here is an example: 私は私立大学で勉強しています。[...]A second year Japanese student could figure this out. For a computer, this is a very difficult problem."
The choice is particularly sad. This isn't difficult at all for a computer, granting you understand the Japanese language. Let's dissect this phrase:
私 は 私立大学 で 勉強しています。
私 わたし (reads watashi) is the English pronoun 'I'. A computer instantly knows it because only the watashi reading/meaning can be followed by the Hiragana は. That's something a 6 years old Japanese knows. And something you would learn in your first months or less of Japanese studies.
Conversely a computer know instantly that's the reading/meaning isn't watashi when it scans that following 私, there's another Kanji. This compound Kanji 私立 reading can only be しりつ (reads shiritsu) out of a staggering number of combined readings of 2 - and that's only because 立 has two usable Chinese reading りゅう and りつ (the third would require the Kanji to be lonesome), 私 only one, し. Kanji have usually one Japanese reading and one or more Chinese readings governed by strict rules on which reading group has to be used. Coding it isn't as much of a headache.
As soon as the computer realize that the third character that follows the first two Kanji is a Kanji as well, the range of possible readings bottoms. That's also due to the fact the first two Kanji makes already a word - as often happens with 2+ long Kanji words, they are compose of multiple words, just like some long words in Western languages would - that means 'private' in English. With the same approach the computer instantly finds the reading of the two Kanji 大学 だいがく (reads daigaku) means University, another very common noun. I think you already got the gist of it. Last word 勉強しています the computer know instantly is a verb because of the unique okurigana しています (read shiteimasu) Present Continuous of "to do" and 勉強 is both extremely popular and has a unique reading, べんきょう (reads benkyou) which is the noun 'study'. "I'm studying at a private university", even a machine translation would be accurate here.

I think the point is that there is no use in sorting all the words written in the three different Japanese alphabets simultaneously in the same juncture. Microsoft knows it so well it has yet to implement it. In your final thought you completely miss to understand that you don't need to attack the problem by "pronunciations". You have only to treat the Kanji with different approach and translate Hiragana/Katakana in romaji, which it has been done already long time ago. I hope at least you're going to quit using "pronunciation" in favor of "reading" by the time you've done reading this post. If ever.