"The twentieth century"
Yes. The suffix -talet can be added to form any century or decade. But "århundrade" is a noun that just means "century".
Literal translation reads as "the 1900's", altho that is inaccurate for English. In English, we say "the 1900's" to mean only the decade from 1900-1909 (in swedish, I think it's "00-talet"?). But in swedish, to say "the 80's" you use "80-talet". Learned this one early on, being an 80's music fan.
Really, your understanding of "the 1900's" is way different from mine. I've always understood The 1900's to refer to the entire century. I've always thought the periods from 1900-1909 and from 1910-1919 did not have a single word/term that referred to each time period, like you would say the 1920's and the 1950's to refer to the periods 1920-1929 and 1950-1959 respectively.
I just looked it up and found that it can be used both ways, see my answer to SN92 below on this page. 1900-talet has the same double meaning in Swedish, personally I'd take it to mean the century unless otherwise stated. (and I'm sure most other Swedes would as well)
Edit: SN92:s comment is below a comment by BerryA1 which has been hidden by too many downvotes so you might have to click on that one to see it.
Native speaker of British English here: until about ten years ago, I had only ever heard "the 1900s" refer to the decade 1900-10. For later decades, we'd say, eg, "the 1920s". Simple and clear. It's only in recent years that "the 1900s" has been used to cover the whole of the last century, which is very confusing and far less accurate!
I understand the spoken difference is mute, but in written Swedish is it wrong to separate a compound noun such as this?
Obviously this is more 'standard' since it's what is being taught, but the question is would a teacher whack me over the side of the head with a fish for separating it?
Yes. It it always wrong to split a compound word.
You might not get whacked with a fish for it, but it's not right.