I was reading a news paper article of Times Of India, and came across a sentence-
To begin with, a woman's right to property has already been established under law. This means that she has equal rights to her parental property as her male siblings. In such a scenario, according women an extra legal right over their husbands' residential property - which too could be inherited - is unfair. Neither do men have the same rights over their wives' property nor can they claim emotional damages during divorce.
Why there is a word "do" in between "Neither" and "men have".?
From my pointing of view It might be - "Neither men have the same rights over their wives' property nor can they claim emotional dames during divorce".
I have also heard people saying that "I do agree with your statement".
Why could not it may be "I agree with your statement".
Is There any grammatical mistake in these sentences or both I can use interchangeably?