1. Big and large are not exactly the same.

When they refer to physical size they are relatively interchangeable. For example, a big house and large house are the same. One is not particularly reserved for height. When we get to metaphorical size, "big" develops more meanings than "large" has. A big man can be large in size or importance; a large man is just large in size.
Why we have two words is because of the way English developed. "Large comes from Latin through French. Big probably comes from Scandanavian.
I understand this.

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2. Dictionaries list "mushmouth" and "mush-mouth". Both are correct, but once the hyphen starts to go, it usually disappears quickly.
I am sorry I don't understand this part "but once the hyphen starts to go, it usually disappears quickly. "

3. Usually you will see "ill-informed" no matter how it is used. "Well informed" is different. We usually only hyphenate words with "well" in a compound adjective if it immediately precedes a noun. "Ill" is treated differently, probably because that use of "ill" is less common.
Could you please explain if you meant I should add a hyphen in the sentence:
She glared at this young doctore so ill-informed about ....
Looking forward to hearing from you.
Thank you in advance.
Jiang
His opinion were well informed.
He had a variety of well-informed opinions.
4. It may have been that and/or the death of her close friends, her spouse, etc.