Don’t talk about my big sister.
Nitpick: the word was not coined in 1900; it’s only the contemporary meaning “brother or sister” that’s first attested in writing around 1900. The OED shows it used around AD 1000 as ‘relative’: “Hæfst þu suna oððe dohtra on þisre byriƷ…oððe æniƷne sibling?”
Tax practitioners tend to call aunts and uncles “non-lineal ancestors” because we very often do need a gender-neutral term for those folks. I don’t think we’re in any danger of that becoming widely accepted. 
True, but it had become completely archaic by 1500 (the OED’s last cite is 1425) and was not part of the language when it was revived. Further, it originally meant any relative, not “brother or sister.” If you’re splitting hairs, then people revived an archaic word and gave it a new meaning, but that’s pretty much the same as coining a word, since all new words (not counting trademarks) are coined from existing ones.
That’s interesting. So if I inquired after your sister, whom I had met in passing, I would say "how’s your ane?
If she was indeed your little sister, would you point out that she was in fact your imouto? Would it depend if it was likely that I would see or hear more of your sister?
Long ago I had a West African neighbor who couldn’t get over that English used the same term – brother-in-law – for your wife’s brother and your sister’s husband, because they seemed like essentially different relationships to him.
Also, your wife’s sister’s husband (your sister-in-law’s husband).
I had a friend tell me that his “brother-in-law” had committed a crime. It took me a while to figure out from the context of the conversation that he was talking about his wife’s sister’s husband (and that was after I guessed incorrectly that he was referring to his wife’s brother).
While strictly speaking it’s not a separate word, in Spanish you can use a diminutive to mean “young” as well as “little;” using an augmentative to mean “old” isn’t so common except for contrast. There have been instances of someone asking my brothers about their “hermanita” and being told “¿hermanita? hey, dude, she’s the eldest: hermanaza! are you calling me old?” (I have 6 and 8 years on the bros)
Is this correct? Because, while I have referred to my siblings’ spouses’ siblings as my “brother/sister-in law”, I always thought that I was using the term too liberally.
You are. In English, your sibling’s spouse’s sibling has no word at all, nor does your spouse’s sibling’s spouse.
I call my brother’s wife’s sister my sister-in-law once removed, but it’s in jest.
I agree with your statement that there is no word for one’s sibling’s spouse’s sibling, but I believe that you are incorrect in stating that there is no word for one’s spouse’s sibling’s spouse. The word for this latter relationship is indeed “brother/sister-in-law.”
The Merriam-Webster online entry for brother-in-law lists the following definitions:
1 : the brother of one’s spouse
2a : the husband of one’s sister
2b : the husband of one’s spouse’s sister
Thai has three words for uncle, two of them gender-neutral. And three words for aunt, 2 of them the gender-neutral words just mentioned. But only one of the words applies in a given case; the choice does not depend on whether you want to be gender-neutral. (This isn’t as strange as it might seem: Thai society is “matrilocal” so one might have a much more important relationship to mother’s siblings than to father’s siblings. And relative age has important social significance.)
“Sibling” is rarely used in English and would often seem a very unusual choice of word, so when discussing a sibling one is more-or-less obligated to reveal the gender, but still we often do not reveal relative age. In Thai the situation is reversed and, since many nicknames are gender-neutral, it’s not unusual to end up mentioning someone’s older or younger sibling without knowing the gender.
Other languages have their own wierdnesses. One (I forget which) uses the same word for reciprocal relations so that “grandparent” and “grandchild” would be the same word, etc.
Actually, this is where the your own/someone else’s comes into play.
お姉さん oneesan: older sister (honorific/respectful, usually used to refer to someone else’s)
姉 ane: older sister (humble, usually used to refer to your own)
妹さん imoutosan: younger sister (honorific/respectful)
妹 imouto: younger sister (humble)
So you’d ask after someone’s oneesan, not their ane.