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
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.