Yes, but that's rarely used. And if used, only for political relations of independence.
In German, selbstverständlich means self-evident, which is sort of like independent. It's a stretch, but I can sort of see a resemblance, and it will be how I try to remember this word.
They've got the word selbständig in German which is the same as självständig in Swedish, so I think you should think of that instead.
Yes, that was my mnemonic too. Unfortunately I got so excited about it that I forgot to translate "work"
why is arbete an ett word but arbetare is an en word. I know people don't control the development of language, but that just kinda seems weird to me
Nouns denoting people are usually common gender, so I think this is one example that actually does make sense, not that our genders generally do.