I'm quite surprised "I own a horse" isn't accepted. Looks more correct (top-level) in English.
Correct. 'Own' is 'besitzen' so the sentence would be 'Ich besitze ein Pferd.'
No, accusative. Pferd is neuter, and neuter nouns take the indirect article "ein" in both the nominative and accusative cases.
If we replace it with a masculine noun, whose indefinite article is different in those two cases, we get:
Ich habe einen Apfel.
Ich = nominative einen Apfel = accusative
So apple is masculine and accusative in most of the cases we've been given so far on Duolingo?
Ich is the subject and therefore in nominative. Den/einen Apfel is the object and therefore in the accusative case.
Der Apfel ist grün. In this case, der Apfel is the subject and is therefore in the nominative case, hence "der". :)
If you hover over the word it will pronounce it for you. It's pretty much like you have phonetically spelled out, but the p is much softer. It doesn't get it's own syllable, but rather is tacked on to the f. It's difficult to explain online, but your lips should pronounce the "pf" in one smooth motion.