There are whole fields of research devoted to the questions you're raising. As such, it's hard to reply with anything that would do justice to them. This isn't to say your questions aren't important, just that your lack of answers reflects your ignorance more so than that of the researchers. I say this not antagonistically but to suggest that it's important to understand that what you see is not always all there is to say.

It is true that these are self-report questionnaires, but as such they are small samples of behaviors of the people in question. Samples of how they perceive themselves, how they think about life, how they think about others, and what they value.

The Big Five, and the measures used in studies such as this, has been validated (in the sense that the ratings have been associated concurrently and predictively) over decades in many ways, with regard to daily reports of behavior, emotion, and life events, diagnoses, work ratings, performance on tests, ratings by peers and colleagues, ratings by strangers, just about everything you can imagine. These self-reports aren't perfect, but they do provide a fuzzy snapshot of someone at a given moment in time. Yes, it would be better to obtain all sorts of other measures of behavior, but they would be too expensive to obtain on large enough samples to be representative.

A major paradox in understanding human behavioral differences is that the more specific and "real world" you get, the less and less they generalize. That is, you can get a very concrete measure of a real-world behavior, but it ceases to be representative of that person across a large number of contexts and situations. Say you want to measure theft, for example. Do you set up a honeypot? Is that representative of that person? Do you use police reports or records? Is that representative? It turns out self-report on online questionnares is a very good measure of things like this because people are less self-conscious, and report things that don't go on the official record.

Faking is also controversial in this area. You're right to bring it up as an issue, but to understand the research on it it's important to think about why someone would fake. That is, what's the motivation for large proportions of people to systematically fake in one direction? And if they do do go to the trouble of doing that, what's "real" and what's "fake"? That is, let's say people make themselves look more dominant than they really are -- what does it mean if one person does that and another does not? It turns out that the person who wants to make themselves look more dominant often is more dominant, all other things equal, because it means they value that.

Also, strangely enough, it turns out that people who are callous and aggressive don't really care about that, especially on online questionnaires, because they are callous and aggressive.

This has all been very thoroughly researched and it turns out to be much more complicated than it seems at first glance. It doesn't mean things can't be better, but it does mean that over very large samples of persons answering questions on a low-stakes questionnaire (in the sense there aren't real consequences to them answering one way or another), a lot of these things average out. It's not the end of the story, but it's not something to be dismissed either.

In the end, questions of sex differences in behavior are about sex differences in behavior. And that's what this research addresses.