Are there any(other than the full complex/1-case)? 
Is there a name for this ($k$-edge-regular I call it)?


Thanks.



Examples for simplicial complexes in which every k-edge is contained in exactly $d$ $k+1$-edges
There are many such examples. If d=2 (plus some connectivity) those are called pseudomanifolds, so there are many of those, and there are many examples for larger values of d. When every set of size k is a k-edge these are designs. 


