In the second game you posted, it looks like White is at a disadvantage since he will lose a tempo after the simple ...dxe4 Qxe4 ...Nf6. That's about what you'd expect from a non-book move with an early queen sortie, sort of like a reversed Scandinavian Defense and Maroczy Bind wrapped into one sad package: not a happy position for White!
Sicilian Alapin's variation- 3. Qc2

You're not going to "refute" Qc2 in the sense of winning the game. You can get a good position by playing any normal moves. In your game after 8 moves (White has just played d4) you have an extremely good French. Compare what you got to 1. e4 e6 2. d4 d5 3. Nd2 Nf6 4. e5 Nfd7 5. Ngf3 c5 6. c3 Nc6. To get from this position to what you had in the game, White would have to play the mysterious 7. Qc2 Be7 and then pass the move to Black. Your real problem is that you had no idea how to play that French-like position.
In bid to improve my play- I have started to analyse my games and make mental notes of mistakes/rare moves made by myself or my opponents.
In 2. c3 Sicilian Alapin's variation- my opponent surprised me with 3. Qc2 move.
Usually Alapin's variation proceed in this manner:
Here how the game with 3. Qc2 went:
A better attempt from black could had been: