As for the second part, it's not that one single proof or another is easier with category theory. But the entire presentation is very elegant. Thinking of Aluffi Chapter 0 where he uses universal property extensively to simplify a lot of proofs. My personal favorite presentation of Algebra, but everyone has an opinion here.
Re: Aluffi, it's a fine book, but I'm not sure I'd teach it that way, because I'm not sure what introducing universal properties and so on really buys you at that level. I guess it make sense if you commit to getting to the end with the abstract perspective on homological algebra. But for a grad student just looking to pass his or her quals? The discussion in Hatcher is concrete, motivated, and involves zero category talk.
Being slightly facetious here as obviously Griffiths and Harris uses sheaves extensively and no category theory.
It's just I see a lot of category hype on here and I want to yell that it's mostly just linguistic! It's not some deep or scary thing. The analogy to matrix notation is really good.
Further, none of the proofs from an undergrad abstract algebra are going to be "easier" with categories. (Show me one?) At the end of the day, you have to prove the same propositions. Categories just give a different and sometimes more convenient language.