Designing an IIR to generate a specific data sequence
Yes, there are time-domain design methods for IIR filters. One of the best-known ones is Prony's method. It is well described in the book Digital Filter Design by T.W. Parks and C.S.Burrus (ch. 7.5).

If the desired sequence is (right-sided) periodic, then there's a very simple and exact solution possible (which is usually an exercise for DSP students).