Those women's lunches . . .
I do not know the context.
It should be "those women's lunches" or "the lunches of those women."
Yes, lunches is the correct spelling.
That is the correct spelling of the plural noun "lunches."
The correct possessive form of "womens" is "women's," with an apostrophe before the s.
'In the hope that' is the correct phrase.
If you are referring to this sentence, no, it does not resemble a correct phrase AT ALL.
the plural for lunch is lunches
This is not a sentence it is a phrase and as a phrase it is correct.
That is the correct spelling of "phrase" (word group, or to use specific words).
The correct phrase is "Did they?".
The correct phrase is "sufficient proof".
It depends on how you use the phrase: Can you provide me a copy of your CV? - correct
No, the correct phrase is veni vidi vici.