Q1: For any given finite simple graph G with e edges, does there always exist an $n$ such that the edges of $K_n$ can be partitioned into $\frac{\binom{n}{2}}{e}$ edge-disjoint copies of $G$? If so, can any upper bounds be placed on the minimal required $n$?

Q2: Similarly for digraphs (with twice as many copies)?

Q3: For any loopless multidigraph with maximum edge multiplicity $m$, it seems clear that for some $M>m$, an $M$-complete multidigraph can be exactly partitioned into isomorphic copies of it, even without increasing $n$: if nothing smaller, one can always overlay $n!$ copies of the original multidigraph with each permutation of the vertices. Can this partition always be accomplished with $M=m$, increasing $n$ instead? If not, can tighter bounds be placed on $M$?

Q4: Does allowing loops (and adjusting edge-counts appropriately) fundamentally change any of the above?