I think in English when 'partner' is used for romantic relationships it is either to be intentionally ambiguous about the gender of the boyfriend girlfriend, but also more commonly for same sex couples. Is that the same in German (although obviously the gender ambiguity wouldn't be feasible).
As you said, in German there's no gender ambiguity (der Partner implies male, die Partnerin implies female, and I'm not sure how genderqueer people fit into this grammatical structure), so using Partner(in) to refer to a romantic partner would only be used to keep some ambiguity about how serious the relationship was - i.e. whether they're just dating, or married for example.
In the context of "mate" meaning "the sexual partner of an animal", yes. Although you can also be more specific and say Sexualpartner when that emphasis is important.
In the context "mate" meaning "friend" in Australian & British English, no. Just use Freund. There's also a German word that carries the colloquial tone of using "mate" in this way: Kumpel. This is specific to guys, though (like "bro").
Die Partnerin didn't work for me in a previous question, when they asked me to translate the partner to German. Is there a reason for this, or was the owl just tired?
Unless there was some other mistake in your answer the version with "Partnerin" probably just wasn't in the system. In such situations you can just report your answer via the report button.
In my college German classes (2008), we were taught that Freund/Freundin each carry 2 meanings: friend - male/female, or boyfriend/girlfriend. There's really no way to tell the difference from context if its a simple introduction, so i can see where Partner/Partnerin are helpful. I personally would introduce my boyfriend as "mein Partner" so there is no confusion as to whether he is simply a platonic male friend.
Yeah, but only because of the possessive pronoun not having the feminine inflection -e, and the noun not having the female suffix -in. You could equally say Meine Partnerin ist stark.
After all, Duolingo likes the sentence Wir sind starke Frauen (although maybe they don't need no man).