Let \((D,d)\) be a metric space, \(CL(D)\) the family of all nonempty closed subsets of \(D\) endowed with the generalized Hausdorff metric \(H\), \(I:D\to D\) and \(T:D\to CL(D)\). In the first part of the paper, the authors study the existence of coincidence points and common fixed points of the pair \((I,T)\). They assume that \(T\) satisfies \(I\)-contractive or \(I\)-nonexpansive type conditions, e.g., \(H(Tx,Ty)\leq k\,\max \{d(Ix,Iy),\delta (Ix,Tx),\delta (Iy,Ty),\frac{1}{2} [\delta (Ix,Ty)+ \delta (Iy,Tx)]\}\) for all \(x,y\in D\) and some \(k\in [0,1)\), where \(\delta \) is the distance function in \(D\). Next, as an application, they obtain some invariant approximation results. In the final part, they present random analogous of these theorems.