"You should apply her ideas."
Translation:Du borde tillämpa hennes idéer.
should have would be borde ha.
bör is an accepted answer here, but the word is not taught in the course. It's pretty formal. The difference in meaning in a sentence like this between borde and bör is very small. (bör would be a little more of a command, whereas borde is a little more of a wish, I think).
No, "Sin" can't mean "His", "Her" or "Its" by itself.
"Hon läser sin bok" means that she is reading her own book, and not someone else's.
"Hon läser hennes bok" means that she is reading someone else's book.
The exception is that it can't be used with "Du"/"Jag"/"Vi"/"Er".
So "Du läser sin bok" isn't grammatically correct.
You would say "Du läser din bok".
Why is "skulle tillämpa" not admitted as correct? It means the same "Skulle" is the past of "ska", which is used as "shall" ("skulle" would be should then). Check http://folkets-lexikon.csc.kth.se/folkets/#lookup&skulle&1&nocorr for reference
skulle is the preterite form of the verb skola, which is most common in its present tense ska. It's a really problematic little arse of a word, one which I frequently claim is the single hardest word in Swedish to teach properly, because it just has so many uses.
The thing is that it absolutely can mean "should", and "must", but only in some situations for which it is hard to come up with examples on the spot. Most of the time, skulle means "was going to", which is the case here - as in "you were going to apply her ideas", likely followed by a "... but [something]".