Are there any fundamental, unchanging differences between male and female brains? Is there any evidence to suggest that stress changes the structure and function of the brain, especially with respect to markers of brain sex?
There are many sex/gender differences in the human brain. (I use sex/gender because we don't know the source of these differences - whether sex or gender.)
The currently known sex/gender differences are not dimorphic. That is, they are found at the group-level, as a difference between the means of the distributions of males and females, but there is an extensive overlap between these distributions. There is no known brain feature in humans for whom we can distinguish between a "male" form (that appears only in males) and a “female” form (that appears only in females). Moreover, people can have both brain features with a form that is more prevalent in females and brain features with a form that is more prevalent in males; that is, brains are mosaics, they are not either "male" or "female". In other words, brains do not have "sex".
Before a race start, competitors' heart rates can reach 180BPM from stress and they report high-stress symptoms such as tunnel vision. Would this be enough stress to alter markers of brain sex?
Such studies have never been done on humans, but animal studies demonstrate that much milder stress can cause detectable changes in brain structure, including reversing the effects of sex on the structure of some brain features. So it is an educated guess that similar effects will be seen in humans. However, it is misleading to talk about changes to brain “sex”. As brains do not come in one of two forms, “male” or “female”, it is meaningless to talk about a change in a brain’s “sex”. What is changing is the specific brain mosaic.
Could regular participation in motor racing during childhood and teenhood teach the necessary skills - spatial reasoning, anticipating on-track competitor behaviour, safely controlling a vehicle - for a career in racing?
Of course, this is called learning! The brain is a very plastic organ, and training, even in adulthood, can alter its structure and function.
Currently, the higher echelons of motorsport are more mentally than physically challenging, with multi-tasking being a primary skill. With this in mind, is there any evidence to support the claim that women are better at multi-tasking, or conversely any to suggest they would struggle?
Some studies point to a gender difference in one direction, others in the other direction; but even if there is a gender difference in this skill, it is a difference at the group level, which is not relevant to the level of performance of an individual. From your description it is clear that for racing one has to be highly capable in multi-tasking, so the only relevant question is: are there highly capable women and men?
Women on average score higher than men in tests of compassion. Would this affect their ability to be competitive?
The same as above, regarding multi-tasking. Even if women are on average more compassionate than men, there are many non-compassionate women and compassionate men.
Men are typically more aggressive than women. Is this due to higher testosterone levels, or intrinsic brain wiring?
We do not know the reason for this sex/gender difference. Remember that also here - there is extensive overlap in levels of physical aggression in females and males, and most males are not aggressive.
Would a female racer with a higher-than-average testosterone level be more aggressive on-track than a female racer with a sex-average testosterone level?
I do not know the data on the relation (if there is one) between testosterone and physical aggression as this is outside my research area. In any case, remember that the link can go both ways. For example, there is evidence that winning a competition increases testosterone levels in humans.
Are women naturally more risk averse? Is their self-preservation instinct stronger than men's, on average?
There are some things in which men are on average more risk taking than women, and many domains in which there are no such differences (for further information, see Janet Hyde's paper from 2005, entitled 'The Gender Similarities Hypothesis'). But as in my answer above, group-level differences are not relevant when talking about the performance of individuals.
Do you think the constant chant of 'women can't race,' the presence and proliferation of grid girls, and the scarcity of women's bathrooms in race paddocks has a negative effect on female racing drivers (much like in math tests, reminding women that they're women by asking them to tick a box indicating gender lowers their scores)?
Of course. This is called stereotype threat, and it means that people do worse when there is a negative stereotype about the performance of their group. For example, American Caucasian men do worse on mathematics tests if they are reminded of being Caucasian (as opposed to east-Asian*). So if there is a strong stereotype that 'women can't race' this stereotype is likely to harm women’s performance in races.
Moreover, it has been shown that increasing one’s awareness to the relevant social group (one’s sex in the case of racing) using simple manipulations like seeing an advertisement, can increase the effects of stereotype threat. So clearly the situation you are describing is highly likely to disrupt women’s performance.
Is there a justification, based on brain sex/gender, to separate women and men in motor racing?
If all the skills that are needed for racing are brain-dependent, there's no reason to separate men and women, even if in the beginning there are less women racers than men racers. Of course changing the anti-women atmosphere is important in order to attract women to this field, and once they are in, to let them reach their potential.