Look no further, I have the answer to all your problems (well only this one).
How to get mouse-free switching between all that good stuff (tips)
I hate whenever I first switch it to Japanese from English (alt + shift), it automatically puts it in direct input.
1. to switch to hiragana from direct input (while in Japanese mode) press ctrl+caps lock
2. to switch to katakana from direct input (again, while in Japanese mode) press alt+caps lock
If you want to type something in katakana, and you're in hiragana mode, F7 is your man!
Hero(hiiroo) -> ひーろー //F7 // ヒーロー
F6-F9 does a variety of things
F7: full width katakana
F8: half width katakana
F9: full width romaji
F10: half width romaji