I don't see how this applies: there are plenty of characters provided by Unicode that can be used regardless of which language I am writing in..
Yes, but they're symbols, not letters (nor ideographs or characters from a syllabary, etc.); and they're even less words. Are you suggesting we replace some pronoun(s) by a symbol? I don't see how that would work. And even if it was adopted in writing, it would inevitably be used in speech, too, and would just as inevitably gain some particular pronounciation in actual language(s).[*]
such as the male/female symbols already mentioned.
But those are symbols; they do not replace the English words “male” and “female” nor their translation in any other language.
So in this case, it would be a symbol for the 'language of the internet', not simply for a single language.
That there be a “language of the Internet” is a strange notion to me. I personally use 6 or 7 seven natural languages to communicate over the Internet, mostly in written form (that does not include HTTP or SMTP ;-) Whatever I read or write correspond to actual words that may be pronounced by speakers of those languages; if I were to use the symbol you would like to invent (and I'm not saying I wouldn't use it, if it existed), it would have to correspond to actual words in the respective languages, otherwise you would just have invented a way of making people mute. In other words, you cannot invent a new symbol if what you want is a new word. They're just two different things. Of course, you may want to invent both at the same time. (I should really have asked for Saussure's _Cours de linguistique générale_ for my birthday as I originally intended. I would have much more insights on the subject. Instead, I got a grammar of Etruscan :-)
Some symbols are available regardless of the general language used, correct?
Of course, you may use any Unicode character you want in your texts. It may make no sense, though.
Granted, I know next to nothing about font encodings, so I'll defer here to the knowledge of others.
This has nothing to do with font encodings, really. It happens at another conceptual level. The issue at hand here is not technical. Arthur [*] It's admittedly a very specific example, but this is exactly what happens with the Hebrew Tetragrammaton, for example (http://en.wikipedia.org/wiki/Tetragrammaton)