-My school gym teacher once tried to tell me that there is literally no difference between boys and girls except for what's between their legs. I have heard similar claims from gender studies classes. That counts as obviously false, surely?

It's wrong on a biological level. From my physiology lecture: Woman blink twice as much as men. The have less water in their bodies.

-People should be allowed to do in their bedroom whatever they want as long as it doesn't harm anyone. Is this contentious? It shouldn't be.

So you are claiming either: "Children are no people" or "Pedophilia should be legal". I don't think any of those claims has societal approval let alone is a clear-cut issue.

But even if you switch the statement to the standard: "Consenting adults should be allowed to do in their bedroom whatever they want as long as it doesn't harm anyone" The phrases consenting (can someone with >1.0 promille alcohol consent?) and harm (emotional harm exists and not going tested for STD's and having unprotected sex has the potential to harm) are open to debate.

-A guy in college tried to convince me that literally any child could be raised to be Mozart. More generally, the whole "blank slate" notion where people claim that genes don't matter at all.

The maximal effect of a strong cognitive intervention might very will bring the average person to Mozart levels. We know relatively little about doing strong intervention to improve human mental performance.

But genes to matter.

-Women should be allowed to apply for the same jobs as men. Surely even people who think that women are less intelligent than men on average should agree with this?

It depends on what roles. If a movie producer casts actors for a specific role, gender usually matters a big deal.

A bit more controversial but I think there are cases where it's useful for men to come together in an environment where they don't have to signal stuff to females.