Short answer is no.
Long answer is everything you do and say will be used against you. If your parents died and you cry out of sadness, of course it's not the end of the world to cry in front of her, but crying because of her or anything deemed 'not deadly serious', you'll be seen as potentially weak. It doesn't make sense but women only need to see strength to feel attraction for you. Now, it is okay to be vulnerable, that's part of what makes you a human being, but telling someone it's okay to be vulnerable is a dangerous road since he'll probably misapply and overuse it, so i tend to tell someone that asks this question no. Seek this security from your male friends/father instead. You'll slip up from time to time in front your girl too, that's not the end of the world.
Regarding the couples who seem happy, well you can't know. They just seem happy from the outside but i can promise you that's often not really the case.