Those women's lunches . . .
I do not know the context.
Yes, lunches is the correct spelling.
That is the correct spelling of the plural noun "lunches."
The correct plural form of "lunch" is "lunches." In English, most nouns form their plural by adding "-es" to the singular form, especially if the noun ends in a consonant followed by "ch," as in this case. Therefore, "lunches" is the grammatically correct plural form of "lunch."
The correct posesive of womens is women's. When used in the beginnng of a sentence it would be capitalized.
If you are referring to this sentence, no, it does not resemble a correct phrase AT ALL.
'In the hope that' is the correct phrase.
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).
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.