Hitting sets with a subfamily
For a constant $d$ the $(k,d)$-hitting set problem is not harder than the original $d$-hitting set (i.e. $k=1$) in view of both approximation and parametrized complexity.
There is a simple reduction from $kd$-HS to $d$-HS. For an instance $(U,\mathcal{F},d,k)$ of the first problem we get an instance of $(U',\mathcal{F'},d)$ of the second one in which every element $e \in U'$ corresponds to a $k$-element subset of $U$, and each set in $\mathcal{F'}$ corresponds to a set in $\mathcal{F}$ in the same way (i.e. mapping all $k$-element subsets of $U$ to elements in $U'$). Since $k$ is a constant the size of new instance is a polynomial function of the size of the first instance ($O(n^k)$).
A hitting set for the first problem corresponds to a hitting set of same cardinality for the second problem and vice versa, hence the reduction is approximation preserving.