Are the "scientific" points made in the James Damore document factually accurate? There has recently been some uproar over a document authored by a google engineer arguing that differences in outcomes between men and women are in part due to biological differences. Are his claims in this area scientifically supported?

His two broad claims seem to be 

(1). Women on average prefer working with people rather than things as compared to men

(2). Women are more neurotic and prone to anxiety
A few thoughts on those two pieces...

Do women prefer working with people?

Results on this are certainly debated. This article summarizes a study (linked in the article) that concludes women prefer collaborative work more than men (though the reasons for this are not necessarily clear). Obviously, standard contextual warnings of interpretation apply. For me, the interesting thing discussed in the article is the potential reasons for these differences (such as male overconfidence). In any case, the answer seems to be "yes, women (on average) prefer collaborative work". Also, at the risk of being mis-interpreted, I should add that individuals do not necessarily follow statistical patterns employed by research to make these claims, and thus, this information should not be used to predict the behavior of a specific individual. It merely discusses certain people's behavior in a certain setting.

Are women are more prone to anxiety?

I don't know much about this, but the first hits on my search (from the Anxiety and Depression Association of America and a The Conversation article from a PhD candidate at Cambridge) both seem to support that women are about twice as likely to have an anxiety disorder. Theories on the reasons for this seem to vary (and not yet be proven), but the first one in both places was difference in brain chemistry. As with the previous disclaimer, this information shouldn't be used to judge individuals. Also, relating to your question, note that this is not necessarily related to job tasks. For instance, the presence of an anxiety disorder does not mean that women are less fit than men to handle the anxiety and pressure of a particular position.

Ultimately, it seems that there may be some scientific support on these claims on average, though: 1) they are not generally "proven" (and may not be able to be proven for the general population), and 2) the results cannot be used to make claims on an individual (this is true of almost all research). Final note is that, in my brief review, the reasons for these phenomenon (if they exist) are primary theorized and not fully studied, casting further doubt upon conclusions for action based upon this research (e.g., women shouldn't be programmers).