There are too many cases here, so I'll just look at one of the most basic: question (1) with $N$ points in general position, conics, and no additional constraints. So any 5 of the points will determine a conic, which misses all the other points (since they are in general position). On the other hand, each point must be in at least two conics. Thus if there are $N$ points and $C$ conics, the number of (conic,point) pairs is at most $5C$ and at least $2N$. The smallest possible number of conics is $\lceil 2N/5 \rceil$. However, this doesn't take into account the fact that no two distinct conics can intersect the same 5 points. Thus for $N=5$ you need 3 conics rather than 2. I don't think this is a problem for any $N > 5$.
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