Stable marriage problem with only one side having preferences
The answer to your first question is: yes, there is a simple augmentation.  It is described in the standard literature on the stable marriage problem.  See the Wikipedia article for references in the literature where this is described.  It is also described here: The stable marriage algorithm with asymmetric arrays.  See also https://cs.stackexchange.com/a/28528/755.

Your second question is an instance of the assignment problem.  We form a bipartite graph, with an edge between a doctor $d$ and a school $s$ if the school $s$ is among $d$'s top 3 choices.  Then we want to know whether there is a perfect matching in this graph.  (If a school has multiple slots, then make multiple copies, one copy per slot.)  There are polynomial-time algorithms to determine whether a bipartite graph has a perfect matching.  See also Existence of bipartite perfect matching.

So, the answer to your second question is: yes, it is possible.