This is my current understanding, I'm not a biologist, so others can step in and correct me if I've made a mistake.

So what I understand is a female produces eggs and a male produces sperm. By that definition, why aren't pre-pubescent kids sexless?

Because already before they're born it's determined what changes their puberty will produce in them. It's not that "some" children will start growing facial hair and "some" start growing breasts, e.g. depending on how "culture encourages them" to do one or the other. What's going to happen in their puberty is clear already when they're embryos in their mother's womb. Of course there might be atypical developments during any stage of development, but biology doesn't work with 100% certainties like that. To give a few examples, there's not an agreed on definition on what's life, or what's a species. Biology is not like Newtonian physics where F=ma.

imagine a woman decides to completely remove all of her genitalia from inside and outside in surgery so that nothing of her genitalia remains, why is it that she does not stop being female

Her genome lacks the Y chromosome that's responsible for starting male development in the embryo. If there's no Y chromosome, the embryo will develop into a female. This genome was responsible for producing her female reproductive system (primary sexual characteristics). You can remove or alter some your primary sexual characteristics (e.g. womb, testicles), or your secondary sexual characteristics (facial hair, large breasts), but the genome that produced all of that will be present in your every cell.

Sex is a biological function in a sexually reproducing species. A species (such as Homo sapiens) cannot be understood on the level of an individual organism and its characteristics. The fact that some individuals in a species cannot or don't reproduce has no bearing on the definition of the species. In fact, loads of individuals in loads of species die without reproducing — they are eaten by predators, die very young, or otherwise perish in the process called natural selection.

And Why are penis, testes, etc male and why are vagina, clitoris, uterus, ovaries, etc female?

Because that's the definition of the word. Well why can't you change the definition, you might ask. Indeed. But this particular definition is based on what has been observed scientifically about sexual reproduction, so it's not that someone just made it up, and that's why it's not justified to change it to what you like. Sexual reproduction came about around a billion years ago, and in sexual reproduction it takes two different kinds of creatures of the same species (in English they're called sexes) to produce fertile offspring. There's only two kinds of gametes, sperm and egg, there's no third kind. There's no "spectrum" there.

Bacteria reproduce by division, many plants can produce by cloning themselves, or by sexual reproduction. Some insects and even some vertebrae can reproduce without sperm fertilizing the egg (it's called parthenogenesis). Just to give examples about various ways of reproduction — mammals reproduce sexually.

Generally it's a mistake to think that there's some one thing that can always decide your sex without exception. Most of the time that is indeed the case, but some people will try to argue otherwise by using very rare outliers as some kind of proof that no generalisations are legitimate. But the more sexual characteristics (and you can start by picking any number of "common sense" characteristics to tell one sex from the other) you look at, and look at them together, the more you will find that they correlate highly. Do you have XX chromosomes, do you have or have you had menstruation, do you have clearly female genitalia, what does your hormone profile look like etc. These are not independent variables, and why would they be because they were produced by the same biological system for a purpose that came to be through the process of evolution. It's irrelevant if you find that in rare cases one ouf ten or twenty of them doesn't match, because you shouldn't look at them in isolation to begin with.

The rare cases do exist, and intersex people are totally real. But they are their own discussion. It's a mistake and evidence that someone doesn't know what they talking about if they think that an exception in biology means "well then you can't say anything about anything".