For the minimum question, this is answered in Theorem 4 of Erdős, Paul; Füredi, Zoltán; Tuza, Zsolt, Saturated (r)-uniform hypergraphs, Discrete Math. 98, No. 2, 95-104 (1991). ZBL0766.05060.
The answer is $\frac{n^2}{k(k-1)^2}+\Theta(n)$, and you can get something a bit more precise from their proof if you need.