Actually, women more or less invented hiragana, and some of the earliest prose fiction, or what we would consider novelistic writing, was written by women entirely in hiragana. It wasn’t that their “poor little female brains [would] explode,” it was mostly a perception of style and gender-appropriateness. Murasaki Shikibu’s Tale of Genji and Sei Shônagon’s Pillow Book are staples of classical Japanese literature today, and were probably recognized in their own time as great writing.
Hiragana are simplified forms of full kanji written in a cursive style, which was originally considered to be a “female” form of writing. It didn’t take long for men to adopt it for informal writing and eventually it became fully mainstream.
Katakana, which are adapted from elements of kanji, were used at first for the same purpose as hiragana, but eventually became a special-use script. Now katakana are used mostly for emphasis, loan words, advertising, scientific or technical terms.
Hiragana are used virtually every other time a syllabary is needed. Since kanji were originally intended for use with Chinese languages, which are definitely a completely different set of languages with virtually no linguistic ties, there were a lot of grammatical markers, verb changes, etc. that weren’t covered by the original writing system. The hiragana you see between kanji in modern Japanese are called okurigana. The little kana you see above or beside kanji characters are pronunciation guides, or sometimes reference notes to make a connection between a translated word and the Japanese equivalent, and are are called furigana.
You could look these terms up if you want details. Way too much to go into in a post, and I’m definitely not expert enough to explain this stuff concisely, clearly, or accurately at a level much higher than what I’ve already laid out.
As far as character entry, computers have made things way easier than they used to be. Japanese typewriters used to have a special layout for kana entry. There was no non-cumbersome way to type kanji, and I don’t believe the Japanese ever used anything like the horribly complicated huge keyboards with a few thousand characters on it that the Chinese did for a while. You can still see the kana layout on modern Japanese keyboards, but if anyone uses it, it’s only the really old farts; everyone else is trained to type with roman character equivalents. Some schools even have keyboarding/wordprocessing competitions for speed.
It really doesn’t take much effort to input anything. You type the roman phonetic equivalent of what you want to say; whack the spacebar until you get to the right set of characters, or pick one from the pulldown menu if it’s a little-used combination; hit enter, and you’re done. It takes me slightly longer to input than English, but it’s definitely not a complicated process. As I said in an earlier thread on the subject, Cecil must have been smoking something.