Learn French: C'est un chien d'origine allemande.

Because the adjective allemande (German) is describing the word origine (origin), not the word chien. Since une origine is a feminine word in French, the adjective must also be in its feminine form.

How does d'origine connect to the question word d'où from this lesson?

D'où means 'from where'. If someone asks D'où vient ce chien ? (Where does this dog come from?), you could answer using d'origine. Notice that both use d' (short for de, meaning 'from' or 'of') to talk about where something comes from.

Why does it say d'origine instead of de origine?

Whenever the word de (of/from) comes right before a word starting with a vowel (like origine), the e is dropped and replaced with an apostrophe. This makes the words flow smoothly together when spoken. The exact same thing happens in d'où (de + où).

Can I just say C'est un chien allemand instead?

Yes, you can! C'est un chien allemand simply means 'It is a German dog.' The phrase in this sentence, C'est un chien d'origine allemande (It is a dog of German origin), is just a slightly more descriptive way to give the same information.