No you don't need to modify the system language, the IME only acts as a layer between your keyboard and what appears on screen. All other text is in your native language and you can switch between typing Spanish and Japanese with keyboard shortcuts or mouse clicks.
So currently I'm using Firefox in French but I can still insert ひらがな, カタカナ and 漢字 however I want

(I just wrote "hiragana", "katakana" and "kanji")
Regarding Android I searched for "writing japanese on Android" and found
this. The app seems to be made by Google but I have no idea if it's good or anything ^^
It may seem a little overwhelming now, and it will take you some hours to set everything up, but don't give up! Once it's set, switching input language is pretty straightforward and you can concentrate on learning
