Hi
I am french native and learnt english at school (for a bit only) and in the states (Boston and New York areas first and Florida then)
So what I would say to answer your question is:
"cot" and "caught": not the same, the o being a short vowel while au is much longer
"Mary," "merry," and "marry": a slight difference I guess, I open my mouth a bit wider to say marry or mary (which are the same) compared with merry.
"hurry" and "furry": the same
"mirror" and "nearer": the same
"father" and "bother": just as mary and merry, I am opening my mouth wider to say the a from father than to say the o from bother.
the "a" in "trap" and the "a" in "bath" Could be the same, but when talking pretty fast I guess the a from bath is not as well pronounced as it should be
