What is the status of evolutionary psychology in academia today?
This is a bit of a quick answer and hopefully others will provide a more comprehensive response.

The number of reflexes that exist in humans is arguably a separate issue to the support for evolutionary psychology. The answer below focuses specifically on the support for evolutionary psychology in academic psychology.

Almost all researchers in psychology would agree that evolutionary processes have shaped what it means to be human. A good example of a well-developed theory is ACT-R, which provides a model of cognition. An underlying assumption of the theory is that the nature of the underlying structure of cognition has evolved to minimise the consumption of computational and biological resources (although you're better off reading ACT-R material directly to understand how they describe the interface of genetic and environmental adaptation, e.g., Anderson 1991). This still leaves many issues regarding how to conceptualise human nature and the extent of human plasticity. 

However, "just so" arguments made in the name of evolutionary psychology are generally not respected in academic psychology. For example, Someone might say that women are better at multitasking because throughout evolution men's behaviour has required singularity of purpose in hunting and mate seeking, whereas women have had to juggle competing tasks of child rearing, looking after the camp, and so on. Of course, that's all rubbish. I just made that argument up out of nothing. Such an argument takes an observation, possibly not even true in the first place, and then seeks to find some vaguely plausible evolutionary story to justify it. It's a "just so" argument. 
You can see further discussion of critiques of evolutionary psychology here.

References


Anderson, J. R. (1991). Is human cognition adaptive. Behavioral and Brain Sciences, 14(3), 471-517. PDF