Therefore, there is probably a different "infinity" point for each pair (-k*G, k*G), which is in accordance with the fact that addition of x-axis symetrical points forms parallel lines.
No, the point is the same. For every public key Q you have nQ=O, where O is the infinity point (0,0). If you have different results, then (as _Counselor mentioned) "you should check your calculations". For two points, where you have d and (-d), your y-value of the public key after addition should be equal to zero.