Learn English – “zh” vs. “j”. Are these pronounced in the same way

I hope I can answer at least part of your question, and with luck perhaps assuage your frustration to some small degree. You asked about origins:

Lose comes to us from Old English

ORIGIN Old English losian [perish, destroy,] also [become unable to find,] from los [loss.]

Choose comes from the same language:

ORIGIN Old English cēosan, of Germanic origin; related to Dutch kiezen.

In Old English, losian would have sounded something like LOH-zee-ahn, while cēosan would have sounded like CHAY-oh-zahn. Both these words are in infinitive form, and would have different conjugations.

In Middle English, losian became losien and cēosan became chesen or chosen. Now, the double-o construction, seen in words like choose and boot, originally indicated a long vowel sound, which itself originally meant literally a long vowel sound, i.e. one that was held for a longer period of time. There weren't any markings to indicate duration, so an extra letter was added to indicate that a word like boot should actually be pronounced the way we pronounce boat today — exactly analogous to German's pronunciation of das Boot, which does not sound like something one wears on one's feet.

But there was another Middle English word for lose, which was leosen (from OE lēosan), and it's not clear if our current word has a single ancestor. Possibly a merging of the two histories resulted in the pronunciation we have today.

Now, I wish I could draw a clear line for you that brings us from past to present and illustrates why today choose and lose and even whose perfectly rhyme but dose and moose do not, and why we pronounce close (meaning near) differently from close (meaning to shut), but the plain truth is I'm just not that smart. English pronunciation is quirky and peculiar in ways that defy description, much less understanding. If there were anything at all to be done about this, we would have an intolerable situation on our hands; but as there is nothing we can do about it, the situation must be endured. Be comforted by the belief that all these pronunciations will change in time — although to what is not at all clear.

I'll leave you with an old joke, and hope you can put aside your frustration long enough to laugh at the pronunciation and spelling mess we have inherited.

Q. How do you spell fish?

A. Ghoti! Just use the gh from rough, the o from women, and the ti from action (or ration or station or — well, you get the idea).

[Source for the above etymologies: Origins: A Short Etymological Dictionary of Modern English, by Eric Partridge]

This interesting page explains that sugar used to be pronounced originally with a common su sound, but (emphases mine):

(...) sometime in the Middle English period the initial letters su shifted to the pronunciation they now have.

If you relax the mouth and tongue somewhat when you are saying the older form, your pronunciation shifts to the modern one, as you’ll realise if you try out the two sounds in turn; the modern version is actually rather easier for slack-jawed English speakers to say. (...) The same change happened with other words, such as sure, and also to words in which the sound occurred in the middle, such as pressure and nation. By the time this shift in pronunciation was taking place, the spelling of the words had already become fixed (...)