Visualization Puzzles #5

All the puzzles come from this position. Black is dominating with a passed a-pawn and more active pieces, but White is going to play Ng1 and Nf3, so it is best not to fool around too much. Thus, I played 35...a4 36. Re1 Qd4 37. Ng1 a3 38. Nf3 Rxf3 (this is winning but completely unnecessary as Black can play ...Qf2 immediately) 39. gxf3 Qf2 40. Rf1 a2 41. Qd1 Rxf1+ 42. Qxf1 Bd4 43. c5 and now

Easy: 41. Qd1? What about the free queen? Why not Rxf2?

Medium: The game continued 43...Kf8 44. c6 Ke7 45. c7 Kd7 46. Qc1 Kc8 47. Qc6 a1=Q+ 48. Kg2. Does Black have a forced mate?

Hard: In the game, I played 48...Qg1+ 49. Kh3 Qf1+ 50. Kh4 Qxf3. Does White have any perpetual check or stalemate tricks?