"A dog and a cat"
Translation:Ein Hund und eine Katze
Why would it be?
einen Hund is the accusative case. You'd use it after prepositions that require the accusative case or when it's the direct object of a verb.
But there's no context here that would require the accusative case. They are just two noun phrases.
So we use the basic form -- which is the nominative case: ein Hund.
Hündin means “she-dog”. If Duolingo suggested it, the most likely reason is that you used the feminine article and wrote “eine Hund” instead of the correct “ein Hund” (with the maskuline article. In cases like these, where DL has a choice of correcting either of two words to match the other, it seems to prefer correcting the second word (in this case “Hund” zu “Hündin” on the basis of preceding “eine”) rather than the other way around.
Why is it eine katze and not ein katze?
Both of those are wrong; it has to be eine Katze (with capital K).
Is a cat a feminine noun?
No; cats are animals, not nouns. But the German word Katze is a feminine noun.
You might think I'm being nit-picky, but you do have to draw a distinction between objects and the words that refer to those objects.
For example, "the center" might be translate as das Zentrum or die Mitte in German. It's not the concept "center" that has a gender; it's the individual words Zentrum and Mitte that express this concept that have a gender. So you can't say that "a center is a neuter noun / is a feminine noun". Grammatical gender applies to words such as Zentrum or Mitte or Katze, not to concepts such as "a center" or "a cat".
Whenever anyone claims that an English word has a specific gender in German, be suspicious.