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At first, the second condition follows from the first by averaging over all $k$-sets containing $a$. At second, under divisibility conditions this construction (Steiner system) exists for large enough $n$, proved recently by P. Keevash. See for example

https://gilkalai.wordpress.com/2014/01/16/amazing-peter-keevash-constructed-general-steiner-systems-and-designs/