The correct answer would certainly be 「買{か}いに」. The sentence would make practically no sense with any of the other three phrase choices.
So we have:
「父{ちち}は今{いま}、新聞{しんぶん}を買{か}いにいっています。」
What does this sentence mean (and imply) exactly? It is saying:
"My father went out some time ago to buy a newspaper and he has not returned yet."
The last part in bold is what is implied by 「いっている」 and it is what the English sentence "My Daddy went to buy a newspaper." does not necessarily imply.
The Japanese sentence above with no further words or context "says" without fail that Daddy has not returned at the time of utterance.
The English sentence feels different in that it can mean daddy has already returned. At the time of utterance, for that matter, this may even be about Daddy's action from weeks or years ago.
This may be a very small difference but my understanding is that SE is a place to discuss subtle differences.
Would it be the same as saying 「父は今、新聞を買いにいきました。」?
This sentence is closer in meaning to the one using 「いっています」 than the English sentence is because it uses 「今」. It is, however, still unclear whether or not Daddy has returned at the time of utterance. It would be completely natural if you said that Japanese sentence after your daddy has returned home.