Depending on the dip angle of the beding (or foliation) planes of the layers and the relation to the slope angle, and if the kinematic conditions are unfavorable the worst case is a failure of the foundation media and complete structure.The most critical from the stability aspect is a case for arch dams. The problems can arise also with stability of reservoar area. The cases for reservoir area of Vajont dam in Italy and Malpaset dam in France are very indicative. The problems with large amount of water loss from reservoir or bellow dam foundations is also possible. So, if the question is for some specific dam profile than the answer by Kennet Towe tells everything.

Your question is more general, yes all the problems you dot down could result. But the magnitude depends on a number of factors/parameters including: amount of dip angle, type of the formation itself, size and type of the dam, relationship of the resultant force/pressure of the dam in relation to bedding/foliation plane, groundwater level and its fluctuation, and so on........for example if you are looking for small earth dam problem of  leakage could be the most critical problem compare to the other two problems you dot down, on the other hand, if you are looking for concrete dam the other problems are also very critical. In summary, the answer depends on many factors (some of them mentioned above). Hope this will help you as a general insight.