Set $N$ points on plane by intersecting as few lines / conics as possible

I have several connected questions. In short form, I want to set N points on plane by intersecting as few lines/circles/conics as possible.

1. Given $N$ points on 2D plane, how many straight lines and/or conics are needed to get all these points as intersections?

2. The same as (1) but there should be no other intersections; that is, exactly $N$ intersections. Possible additional constraints for special cases:

   • only straight lines allowed;
   • only circles allowed;
   • only conics are allowed;
   • there are no three collinear points;
   • there are no four concircular points;
   • there are no six points that are on the same conic (BTW, is this true that any five points on plane will determine a conic?);
   • maybe something else?

Thanks for any help!