Only on average, and it depends on what type of strength you are talking about. There is actually a lot of overlap for certain strength-based activities, such as distance running, carrying heavy-loads, and walking. Men, on average, do have higher grip-strength and upper body strength.
No, not exactly. On average, men hunt and women gather, but in some hunter-gatherer societies there is far less gendered roles for these two activities, so one can't draw up tidy little dichotomies. Futher, some types of hunting doesn't involve or rely on men's strength; in fact in more than a few hunting-gathering societies men hunt large game not because they are strong, but more likely because successful hunters are viewed as more prestigious and can acquire more resources (and females) by being viewed as a successful hunter. Studies have suggested that hunter-gatherer men would actually obtain more animal protein if they hunt/trapped small game (something that doesn't require strength), but they don't do this; instead, they pursue risky animal prey for status reasons. Certainly, from what I understand, strength plays a role in some types of hunting, but I don't think hunting was the evolutionary reason for sexual dimorphism in body strength.
I'm not aware of any evidence suggesting that "bigger & stronger" will lead to an instinctive "responsibility for bigger & stronger work"
Again, these "instincts" you refer to depend critically on our ancestral social/mating system, and it is unclear how territorial/competitive our ancestors were, so it is unclear exactly how territorial/competitive present-day humans are supposed to be. If anything, several psychological studies suggest humans will rapidly readjust their in-group/out-group competitive mentality in the face of a larger threat (A hates B, but A and B unite against common enemy C), indicating that these are not hard-wired and fixed instincts, but flexible psychological "programs" that are capable of responding to external stimuli. And one of the largest external stimuli out there is culture. If you grow up watching/experiencing violence (a cultural phenomenon), you will end up being violent. Same goes for non-violence and its influence on your demeanor. Compare, for example, the !Kung versus the Yanomamo. The former is quite egalitarian, passive, and monogamous, the latter is quite fierce and polygynous.
Women tend to want to have children more than men, on average. But I'm not sure this has much to do with women impelling themselves to working less hard than men, which is what you wrote. That seems a rather harsh and unsupportable judgment on women's work ethic, particularly when there are so many other real obstacles in their way such as sexual harassment, to name one. Raising kids while holding down a job in a culture that is predominantly male-oriented is another.
Also, women (and females throughout the animal kingdom) are highly competitive, just not competitive in a way that a right-leaning man such as yourself might see them.
It is a common and false dichotomy to talk about "biology ends and social/culture begins" They interact, and as I said before, our brains are not hardwired with fixed instincts, but are wired up by evolution to be capable of adapting/adjusting to external phenomena. We don't have an hard-wired and adapted brain; we have a brain capable of adapting.
I'm not aware of masses of students with aimless social-studies degrees not getting on in the workplace...where is this idea coming from?