Does the existence of intersex people invalidate the binary conception of gender?

You ask a complicated question very simply! Here's some advice about how to pursue this topic, with a few oversimplifications of my own. Sex (physical sex) is often distinguished from gender (gender identification in people, often culturally influenced) as well as from sexuality (sexual orientation). Intersex people have bodies that are not "typically male bodies" or "typically female bodies" but have elements of both. This ranges from (controversially) hypospadias in men (the urethra not opening at the tip of the penis) to individuals with both an ovary and a testis. Anne Fausto-Sterling's excellent book Sexing the Body describes the range. Then the question is, do we regard intersex individuals as "abnormalities," and thereby preserve our traditional understanding of biological sex as a binary, or do we regard intersex individuals as counterexamples to our traditional understanding of biological sex? Some (including Fausto-Sterling) appear to think that the answer to this depends at least...

People suppose that hard science is more objective than other subjects such as psychology. But doesn't science require good instincts, judgment, and intuition like any other field does? People say well all the scientists agree that global warming will have a big impact on the world but how can I really be so sure that it's as simple as "science sees it some way so it must be correct?" Isn't it just an unfounded prejudice that scientific judgements can be validated in some essentially simple and uncomplicated way?

You are asking a few questions here. One is whether you should take it on trust (or authority) that scientists are in agreement on a scientific questions such as global warming. Another is whether or not assessing scientific evidence is "simple" (and I think you are right in suggesting that it is not simple). And a third is whether or not science is "objective" (a complicated question that philosophers of science, as well as scientists, often debate). A final question you may be asking is whether physics or climate science is more "objective" or "simpler" than psychology or other social sciences, again a complicated question that there is no general agreement on.

Are there any non-academic journals that someone who isn't a professional philosopher with a degree in philosophy or affiliated with a university could publish a rigorous philosophical paper?

If you have a rigorous philosophy paper, why not send it to an academic journal? You do not need to have a PhD in philosophy or a job as a philosopher to submit to a journal. Best to look for a journal that does "masked" ("blinded") review, so that your identity is not known to the referees and they won't be biased by your lack of professional stature.

Sometimes people will try to discredit the validity of a scientific experiment by saying that the results don't apply to the real world. Is that a valid argument?

It is a good argument only when there is reason to think that the experimental situation may be different in some relevant ways from the natural situation. So, for example, tests of nuclear bombs in desert areas or underground yield results that DO apply to the real world. Tests of drugs in vitro (in the test tube) may not apply in the "real world" of living organisms (in vivo). Nancy Cartwright is a philosopher of science who has written extensively about these issues.

If in the future, science makes it possible to use cloning to "create" Neanderthals which were isolated in their own environment, would the revived species of Neanderthals evolve back into Homo Sapiens millions of years later? Would the process of evolution yield a new species of "humans" with Neanderthal ancestry?

You ask an interesting question about the process of evolution. Neo-Darwinians typically argue that species evolve by natural selection on random mutations. At any particular time, there is more than one kind of random mutation that is of selective advantage, and it is contingent which one of these (if any) occurs. So evolutionary history has a certain randomness and unrepeatability. If we recreated Neandertals and did not interbreed with them, they might evolve into yet another (new) human subspecies. Some biologists have argued that some mutations are directed--i.e. not random and in response to specific environmental challenges. To the degree that this is so (and that environmental challenges repeat themselves) evolution might repeat itself. But this is a controversial theory and is not generally thought to account for much mutation.

Does science have its own built in "selection bias" toward things that are measurable or relatively more measurable?

Since the Scientific Revolution, scientists have valued the combination of natural science and mathematics. Quantification (measurement) is valued in part because it contributes to precision in making predictions or interventions. The more precise a prediction that is made, the more confirmed a theory is if it the prediction is verified. That said, sometimes the preference for using numbers is valued in itself, or for the aesthetic pleasure it provides some people. And I think you are right to suggest that this may be a "bias" in that it may lead to devaluing sciences that are not, and perhaps cannot be, quantitative.

If a person does not believe P, where P is some proposition, is it fair to say that they then positively believe not-P?

Here's an example to help make the question concrete: Let P be "It is raining." I don't believe that it is raining. However it does not follow that I believe that it is not raining. (I don't have any beliefs about raining at the moment.)

Here's an example to help make the question concrete: Let P be "It is raining." I don't believe that it is raining. However it does not follow that I believe that it is not raining. (I don't have any beliefs about raining at the moment.)

We often deride others by referring to them as childish. Why is this an insult? What's so bad about being a child? The only major disadvantages of being a child I can think of are physiological and intellectual, and yet when we say someone is acting childish, we usually don't mean they can't perform complex reasoning or that they haven't reached the peak of their physical prowess. So how are we to understand accusations of childishness as insults?

Children have poor impulse control. They often think the world revolves around them. In general they have difficulty managing their feelings and often end up in meltdown or, worse, in violence. These emotional aspects of childishness are much of what we have in mind when we say that adults are acting childishly. And our main goal is not to insult people but to remind them that in the process of growing up they (should) have learned skills that they should be using now.

How reasonable is the way we speak about causality? Say a person catches a cold. The cause of that cold might be said to be the effect of the cold virus; or it might be said to be the contraction of the cold; or the failure to prevent the contraction of the cold; or the presence of the virus or of the victim wherever it was contracter; or whatever brought either of them to that place; etc. For most things (leaving aside the thorny issue of free will), things that happen are caused by other things. So when we speak of causality, does it make any sense to say that some causes caused whatever we're talking about, and to ignore other, more proximal or more distal causes?

Usually when we ask a question about what caused something, we are engaged in trying to repeat or avoid the same situation, or trying to assign blame and responsibility. So although events have many causes, only a few or one of them may be relevant in a particular context. If I disregard instructions to quarantine myself and infect you with the TB bacillus, then one of the things we might say is that the cause of your getting TB is my irresponsibility. (It would not be helpful to say that the cause of your getting TB was the TB bacillus, or even that the cause was my cough.)

A colleague of mine is a very devoted vegan. So devoted, in fact, that he argues that it is morally wrong to wear fake fur or fake leather, or to eat any kind of non-meat food that is meant to look or taste like meat. Apparently, doing so symbolically condones tyranny over animals, supports the meat and animal-based fashion industries, and demonstrates disrespect and contempt towards animals. Now, I have nothing against veganism, but this just seems too radical. Is this kind of argumentation sound? Or are there any more sensible arguments against fake fur or leather, or meat-like food items?

The claim that something "symbolically supports tyranny..." is not a claim about the act in itself, but a claim about the meaning of the act. Your vegan friend may see troublesome meanings in the act of eating artifical bacon flavored chips or wearing fake fur. But not everyone does, and there is much ambiguity and complexity about what things mean. To take another example, I have friends who will not marry because they think that "marrying symbolically supports an institution that oppresses women." I don't doubt that marriage has historically oppressed women, but I think marriage has multiple meanings (commitment, family, for example) and any decision whether to marry or not needs to take all these meanings into account. Back to the fake fur: Personally I prefer fake fur that looks fake, so that I don't make unnecessary enemies or set a bad example. Your friend is so passionate about veganism that he focuses on one set of meanings.