I think this question is a little bit confused. First of all, there is a phenomenon in some varieties of American English called the Mary-marry–merry merger, where all or some of the words Mary, marry, and merry are pronounced the same; that is, the vowels /e/, /æ/, and /ɛ/ are all merged before the sound /r/. In some varieties of American English, there is no merger, and all of the words merry, marry, and Mary are pronounced differently. In some other varieties, two of them are pronounced the same (which two depends on which variety). In other varieties, all three are merged and pronounced the same. There is a discussion of the merger and an associated map in The atlas of North American English, page 56.
As for the names spelled Sarah and Sara, as far as I know, in all dialects of American English, they are pronounced the same, regardless of whether Mary, marry, and merry are pronounced differently.
For reference, in my own speech, I pronounce them all the same (in the way that those who pronounce them differently say Mary), but I can produce the three-way distinction if necessary—usually only to demonstrate this phenomenon.
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
Interestingly, this question appeared as number 15 on the Harvard Dialect Survey, so it is possible to give a good summary of the pronunciation differences in these three words as they are spoken in the United States.
The 11,422 respondents were asked to choose from five options given the following prompt:
The percentages next to each choice indicate the proportion of survey participants that chose that option:
The maps that show where in the US the respondents for each answer choice were located are available here, but can be summarized by noting that mid-westerners (residents of Ohio, Illinois, etc.) seem most likely to pronounce all three the same, and New Englanders seem most likely to pronounce at least two of them differently.
As for non-US speakers, a linguistics blog post by Ryan Denzer-King claims that,
So it is more likely that you would hear people talk without these three sounds being merged into one in places outside of North America. This seems logical to me, but I don't know enough rhotic speakers to say if it matches my experience, and there will of course be exceptions everywhere.