Are all problems in the same time hierarchy related to each other?
If problem A is complete for the class, the class is higher than P, and problem B cannot be solved in polynomial time, then yes, there is some polynomial algorithm that makes use of the oracle and solves the problem.
By definition, if a problem A is complete for some class C, then every problem of this class is reducible in polynomial time to this problem. As a result, a polynomial algorithm for a problem B given an oracle P of the (complete for the class) problem A (we assume that the oracle spits out the answer to an instance of A in constant time) is the following:
Input: instance of problem B

Reduce the instance of B to an instance of A (polytime)
Answer the obtained instance of A using P (constant time)

However, it is not always easy to find the reduction from one problem to another in the same class.