If you're married, no. Marriage is a partnership. If you're single, then yes. Why is it so important to be head of household? Your spouse is your partner, just because he doesn't do the paperwork doesn't make him any less an equal partner.
Yes it does. If the woman is the sole provider she is head of household, primarily speaking about tax forms. Even if she's married, if she is the one handling all financial aspects, she gets the title.