The answer is historical. In Old English, voiced and voiceless fricatives were phonologically equivalent. (The current English graphemes <v> and <z> did not exist in Old English.) Between voiced segments, voiceless fricatives became voiced--and this is called sibilant softening. Then, in Middle English, these fricatives became phonemic. That's why you see these discrepancies. Words with <s> between voiced segments after the Middle Ages don't fit in this pattern. That's why lousy (from the 14th cy) got /z/ (softened /s/), whereas mousy (from 1853) got /s/.
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]
Best Answer
Copper comes from Old English, where it was spelled coper. It was probably spelled with two p's in Early Modern English because it was a short vowel, and there was a tendency to double consonants after a short vowel.
Proper comes from French propre, and before then from Latin proprius. It was probably spelled with one p in Early Modern English because there was a tendency to stick closely to Latin spellings (e.g. debt, which never was pronounced with a /b/ in English, but which had the 'b' added in Early Modern English because it originally came from Latin debitum.)
As the comments noted, all of the others come from a one-syllable word with the suffix -er added. The rule (simplified) is to double the consonant after a short vowel when you add the suffix -er.