Oh gee guys, I've got a lot to respond to here. :-p
Roy:
As has been mentioned here, "D minor" is a key in which a piece is based. That means the the inherent structure of a piece follows a certain chordal progression (sometimes), and speaks to the intervals between notes. In D minor, there is one flat, which is a B flat. Confusing, perhaps, that the key with the B flat is called D minor, rather than B flat minor or major or whatever, you know. We musicians like to be confusing, often for the sake of being confusing. :-p
The key of a piece does indeed speak to the "mood" of the piece. As a general rule, "major" keys are described as more upbeat and jovial, whereas the "minor" keys are more serious, serene, and melancholy. In my experience, pieces in B major are trash :-p Well, I just don't usually care for that key. In terms of major keys, A, D, G, and C major are my favorites. For minor keys, I really like E and D minor. This may speak to a concept that Byron mentioned. I play violin. In terms of fingerings, those keys are, in my opinion, much easier to finger. Hence, with better fingerings, violinists get better intonation, and therefore I may consider an in-tune A major piece to be "better sounding," with its three sharps, than a piece in D# minor played out of tune, with its extremely formidable six sharps. A key signature with six sharps basically is asking one to take notes (like in a lecture) in English given by a professor in German -- it's certainly possible with a great deal of familiarity, but it can be one major task to immediately "translate."
Don't let the whole "major," "minor" thing throw you off. It has nothing to do with the pieces' significance.
Opus 13 refers to a catalog of a composer's work. As David pointed out, it typically refers to the actual order of publication of a work. I have a funny story about those symphonies. Dvořák worked with various publishers over his career. My standard practice when I get a piece to work on in orchestra is to get a good recording and listen to it extensively. So in the fall of last year, we the orchestra got the Dvořák 8th symphony music to sightread and eventually perform. So I went home and bought a recording of the symphony online. Unfortunately, we were using an older edition from way back towards the turn of the last century. At that point, the publisher deemed Dvořák's 8th symphony as the 4th, because that's how many symphonies Dvořák published with that specific company. So, I accidentally discovered the 4th symphony, and I absolutely love it.
"From the New World" is simply a title given to the symphony. On occasion, or even by regular practice, pieces can be "nicknamed." I believe Dvořák had recently emigrated from some Czech territory over to America when he wrote the 9th symphony. There is disagreement on the issue, but supposedly he derived basic melodic and harmonic structure in especially the last three movements of the piece from the "essence" of the "New World." He considered that to be African-American spirituals and Native American music, supposedly. Whatever the case may be, it is absolutely glorious and beautiful music. I've had the opportunity to play 9th symphony with both the Erie Philharmonic and a state orchestra festival -- imagine, if you get a chance to listen, being engulfed by an orchestra of over 150 musicians, in the 4th movement of the 9th. It was so miraculously energetic!!
In any case, pieces do sometimes have meaningful titles like that. Haydn was apparently a big fan of assigning names to his symphonies, and Mozart did it on occasion, as do many composers.
I'm sure that Bob Dylan takes at least some influence from the great composers of the last few centuries. That might explain why his music is so great, too :-p .
Roy and yankeerunner, don't hesitate to ask questions like that! I'm actually thrilled that anybody gives a damn about such music! In fact, keep asking questions about other stuff too. It's healthy.
Byron:
I'm actually a fan of the Mozart piano and violin sonatas. Hilary Hahn and Natalie Zhu produced some impeccable recordings of some of them. In fact, I also thoroughly enjoy some of Mozart's concerti (bassoon, horn, and violin, especially) and serenades, although not so much his symphonies. I'm also a huge fan of Rachmaninoff. Yet, I feel that largely, the music of both great composers still remains "trite." That is, I feel it is overly repetitive, largely unoriginal, and generally just not that exciting. Of course, such sentiments on music rely heavily on just personal opinion. Even music I, in my great qualification as a prolific composer and international musician superstar... or something... deem as "trite," I usually still do enjoy. I just don't think that Mozart's music was, in context, very extraordinary. I would hardly describe it as "sublime." In fact, I'd say quite the opposite. In an unjust oversimplification, a large consideration with Classical era music was to produce a great deal of highly-structure music. That is, music in rondo or sonata allegro form, most typically. In terms of Classical era composers, I have a pretty decent view of things given my exposure thus far: Haydn defined the Classical era, Beethoven ingeniously expanded music out of that era, and Mozart just worked within the framework provided for him. Excuse the oversimplification -- there are so many, many many many many great composers that did so much to define music in that era and evolve out of it. I say that not to devalue Mozart's music -- I enjoy it. That's why his music is so popular today -- he found what is inherently pleasing, and exploited it, writing hundreds of pieces that are, given developmental changes, subject to the same concept of what music is. So, he's prolific and popular now. I just feel that, for example, Beethoven's string quartets, even Haydn's quartets, surpass Mozart's in any number of ways. I could write a whole paper on such topics -- in fact I have... which was pretty intense... but of course I usually don't have any idea what I'm doing or saying, so I wouldn't pay much attention to me anyway. :-p
You know, as I'm writing this, I'm listening to the Mozart piano and violin sonata in G, K.301. Trite, but pleasing.

Oh, there's another note: some "special" composers have their own catalog of works. Mozart, for example, doesn't have works categorized as "opus," but rather as "Köchel," or "K." Bach is BWV, f.y.i.
Schubert -- ew, nasty. Just kidding! :-p It's almost cliche, at least for voice people, but I love Schubert's Erlkönig, D. 328, Op.1: Wer reitet so spät! Get it! Listen to it! Love it!!! It's definitely intense, and workout-worthy. Go! What are you waiting for!?
Please, I welcome any other inquiry, even if I may not be able to answer it. I love music, and any discussion on it is valuable!
Phil