We actively try to not fall into gender stereotypes with the course sentences. :)

'henne' is the direct object form, meaning 'her' as in: Han älskar henne (He loves her)

but 'hennes' is the possessive form, meaning 'her' or 'hers' as in: Kjolen är inte hennes (The skirt is not hers) Hennes bil är röd (Her car is red)

Yes, it's a common word, but it might be the case that Swedish uses it more generally in English. My colleagues at work have the title "medicinsk sekreterare", but they're not anybody's personal secretary, rather doing administrative healthcare work.

I am in the States and think of secretary as someone who writes memos, prints and get papers, takes calls, and schedule meetings. While an assistant may do these things, they will get more job experience by doing small projects, writing research papers, etc. I would also consider secretary as more generalised, while assistants can be administrative, legal, lab, etc.