Your title "Problems in Evolutionary Psychology" is a bit ambiguous. Does it mean problems with the field of evolutionary psychology, or does it mean problems in the field of evolutionary psychology that evolutionary psychologists themselves are grappling with?

This post introduces some of the issues in studying evolutionary psychology, but off the top of my head, my reading on the subject suggests that evolutionary psychologists are aware of most of them and have taken them into account. So the second meaning would make more sense to me.

Either way, I think it would be useful to discuss what evolutionary psychologists think are the inherent difficulties in their fields (and how they propose to deal with them), and how they answer broad criticisms directed at the field. Evolutionary psychologists have been criticized a lot, and have spilled gallons of ink explaining their views in journals. I will try to find something that's a good introduction and that isn't behind a paywall.

In particular, evolutionary theories about sex differences seem to get mentioned and appealed to as if they had an iron-cast certainty.

Could you give an example of someone making this error?

Women have less strength than men and are the ones who bear children, which could easily have affected their social position even without drastic psychological differences.

Ah, but why do women have less strength, and men have more? See the excerpts from David Geary's Male, Female here arguing that greater male strength is related to sexual selection. (The mere fact that females have the babies isn't enough, because many monogamous primates exhibit minimal dimorphism.)

We know that there were different selection pressures on men and women. It doesn't make sense to believe that these selection pressures were strong enough to change body morphology, but somehow had no effect on psychology and behavior. That would be "neck-down Darwinism."

Of course, it is hard to tell whether a present-day sex difference is an adaptation, or not. A lot of arguments that X is or isn't an adaptation seem too black-and-white; instead, we should be talking about probability that a trait is an adaptation. In many cases, evolutionary psychology has made predictions based on hypothesized adaptations, which have turned out to be true, especially in the area of mating preferences. I'll try to get some citations for you. These findings should lead us to increase our probability estimates that such behaviors are related to adaptations.

Occasionally, the studies purporting to show cross-cultural sex differences actually show that the differences are smaller in the more egalitarian countries.

I think your post could use a couple citations for this claim. Off the top of my head, this claim may be true for some traits, but I've also seen evidence that it is false for others.

Even behaviors that were very widespread may or may not apply to any particular individual. Lists of ”human universals” will tell us that members in every tribe found so far will interpret facial expressions, love their children, tell stories, feel pain, experience emotions, and so on. But there are also individuals who do not know how to read facial expressions, do not care for their children, are not interested in stories, do not experience pain or emotions, and so on. Sexuality is one of the drives that would have had the strongest selection pressures operating on it, but we regardless have people who have no interest in sex, are mainly interested in sex with things that you cannot reproduce with (same-sex partners, children, cars...), or prefer to just masturbate.

It's true that there is variation. Though variation needs to be investigated further before it tells us much about the plausibility of evolutionary hypotheses. Evolutionary accounts have many reasons to predict variation, e.g. frequency-dependent selection.

Furthermore, some of the examples you describe may be related to biological but non-evolutionary factors. There is evidence that prenatal hormones influence human psychology. Many factors can influence the prenatal environment, such as maternal stress.

So, just because we see a certain sort of variation, it doesn't necessarily strike down the hypothesis of universal, or quasi-universal, evolved human predispositions. Prenatal factors can counter-act or modify evolved predispositions. Of course, we shouldn't hand-wave any variation by saying "that's just prenatal noise;" we need actual evidence suggesting that variation in a trait is prenatal. And for some items on your list, that evidence exists, so that variation doesn't cast much doubt on the plausibility of a high evolved similarity in human psychology.

That sort of information can only be found by ordinary empirical research,

Certainly. But don't evolutionary psychologists know this? And I'm talking about what evolutionary psychologists write in peer-reviewed publications, not speculation in popular books.

and ordinary empirical research doesn't need evolutionary psychology for anything else than suggesting interesting hypotheses

No, but your language here seems a bit strange, because a method for generating interesting hypothesis is a really important part of science. Having great procedures for testing ideas is no use if you have no ideas.

To me, that quote sounds something like saying "boats don't need navigators for anything else than finding where they are going at sea."