I just checkmated my first 1678 player:

My rating fluctuated from 1450 down to 1395 (series of strong 1250 players got me) and then climbed to a steady 1500. I'm now on a winning streak against 1350 players, who are my most common opponents.

This game took 2 minutes each side, even though we both had 10:

Did he just play weakly or am I getting good?

I think it's hard to tell whether you're getting better just by looking at single games, especially fast ones where blunders are more frequent.  Even winning or losing half a dozen games in a row isn't really a good indicator, since practically everyone experiences temporary streaks of both kinds.  The real litmus test lies in being able to sustain a level of play over a protracted period.  In other words, the question isn't so much whether you can reach 1500 or not; it's whether you can sustain that level (more or less) over the next 50 or 100 games.    

In this particular game, as black I probably would have done 15...Bf5, which I suspect would have made your attack much less straightforward.  On the other hand, I'm a long way from being a grandmaster.

5. ... h6 creates a weakness. As black I prefer to wait for Ng5 and then play what my computer plays: ...Nh6, where the g5 knight blocks the c1 bishop from taking on h6.
6. ... exd4 is a mistake, giving up the center to white. Better is Bb6, leaving my c pawn on c3 where it blocks my knight. 7. d5 closes the center, and dxe5 will lead to Black having only a slight set back in time.
8. Knights before bishops. My plan is Be3, but I don't want to scare black from castling kingside just yet.
9. I can cramp black's position temporarily with e5, driving back the knight, but it would not last long. That move should be saved for right before an attack. In chess you want to set up a series of blows, but not unveil them until your opponent would not have time to respond to a double attack.
9. ... 0-0? That side of the board is a bad place to castle. I smelt blood when that move was made. Note how the dark squared bishop is stuck on the wrong side of the board.
10. Qd2. Any 1400+ player should know what that move is for, as this player did. But Ng4 fails to h3. Kh8 followed by Rg8, or g6 followed by Re8, or Nh5 are worth considering.
13. ... Rg8 is far better than gxh6.
14. ... Kg8 is stronger than Nh7.
15. ... Bf5 can save black
17. ... Rf7


Black had tons of opportunities to ward off the attack. I think the owner of this account let a much weaker player use it.

Yes, my attack was pre-mature. I've used a variation of this attack in 75% of my last 20 games, and most players rated lower than me fall for it. At worse I'm down a piece for two pawns and maybe lose a few tempo, which can be recovered from. I have lost a few games though against people who knew how to defend and then hang on to the extra pawn till the end.

Usually I develop while giving a few threats to slow down my opponent's development, and I try to keep a hold in the center and ward of my opponent's threats. But once developed, I don't know exactly what to do, and often I go for one of these sacs to see if my opponent falls for it. 80% of the time they do. I guess I should be learning more ways to strengthen my position instead.