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The left-brain/right-brain thing is a myth.

“there is no scientific evidence that individuals use one half of their brain more than the other or that an imbalance in the two sides explains individuality”

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The left right brain idea is not only a myth as u/BlackHoleHalibut points out, but evidence has shown pretty paltry (if applicable) difference between women and men skillwise (in practice) and biology-wise. Things that were accepted as the norm back then (ie women being bad at math) were shown in later years as more women got into university were found to be negligible. Environment plays a great role in the manifestation of certain traits - socialization (it's generally accepted or expected women to be more social or emotional, while boys are taught to sublimate their emotions in different ways), something called the self-fulfilling prophecy or an expectation coming true cause the people it pertains to make it happen in part cause they are convinced it's inevitable. That isn't to say there aren't cases where gender presentation is different. Schizophrenia tends to manifest less severely in women and later in life - though in part due to the protective factor (something that helps manage a trauma or disease) that is estrogen. Women have been documented more at risk of developing Alzheimer's but the reasoning is still being researched.

Nope. The biggest difference is that male brains tend to be a little larger than female brains (which tracks with body size). Aside from that, there are essentially no differences between male and female brains. It's also important to note that brain size has a very weak correlation to intelligence so its not possible to make any claims based on size alone.

Human brains are not dimorphic - www.sciencedirect.com/science/article/pii/S0149763421000804

Brain mass and intelligence - www.scientificamerican.com/article/does-brain-size-matter1

No that is an excuse people use for sexism, and if anything women tend to be slightly more logical behavior wise (less risk taking/less likely to engage in risky behavior like drugs), not because of a difference in brain activity though.

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Right? I never had to write up any of my female employees for filling their sinuses with grease on a $5 bet, or trying to pin casters to coworkers in order to engage in bizarre "tech races" in which they intended to sit on the be-castered coworker and steer them via a length of nasty rope from the pit in the fashion of an equestrian bit.

While the degree to which the brain is lateralized is often exaggerated, the other responses to your question are downplaying it too much. One of the most significant aspects of lateralization is in areas of the cortex that regulated language processing. It is generally the case that in most people, the left cortex processes the rules and structure of language, whereas the the right side of the brain processes prosody, which is emotional intonation of language. This lateralization is so pronounced that it manifests as one the most distinct forms of lateralization in the brain, anatomically speaking. Specifically, the planum temporale is in some individuals as much as ten times larger on the left than on the right. This difference was first described in seminal work by Wada et al in 1975 in JAMA.

The degree to which this lateralization exists differs between men and women is not entirely clear. Wada et al and other studies found a sex difference, suggesting that men are more likely to be lateralized than women (i.e. left larger than right), which is part of where this idea that women are more "emotionally centered" in their brains than men. Other studies have not found this difference, however.

A fairly recent study did find a sex difference. Interestingly, they also in men with Klinefelter's syndrome, wherein men have XXY sex chromosomes. These men (I need to be careful here, because some studies have shown the individuals with Klinefelter's syndrome are 10X more likely to be transgender than XY individuals) tend to have female-like breast development and other feminine features. The study above also found that XXY men were not different from women in the degree to which their brains are lateralized.

Anyway, there are other studies showing functional differences in emotional processing of faces between men and women, specifically related to lateralization in the brain. In general, males tend to have a stronger response in the right amygdala than in the left.

The final point I will make is that there is enormous overlap between men's and women's brains. We can find differences in lateralization, size of various things, etc, but the differences are relatively small and not always apparent in every study. Further, you cannot take any one individual brain and make clear judgments about how "male" or "female" it is necessarily. Exceptions always exist.