Are there problems for which divide-and-conquer / recursion is provably useless?
Is there a problem with an exponential algorithm which has no algorithm with polynomial recursive runtime?


Yes.  Note that if a tally language has “recursive algorithm” with polynomial “recursive runtime,” then it is in P.  There is a tally language in E∖P by a standard diagonalization argument.


  Is there a problem which can be solved in time $f(n)$, but provably has no recursive algorithm with recursive runtime $o(f(n))$?


It may depend on the computational model, but I doubt this is known, given that the time hierarchy theorem for two-tape Turing machines is not powerful enough to distinguish, say, O(n2) time and o(n2) time even without giving the latter the constant-time access to the answers on smaller instances.



Random comments on your questions in this post:


I do not think that these questions are related to the usefulness of recursive programs, despite what the title of the post and your choice of terminology suggest.  As BVMR wrote in an answer, recursions and iterations are equivalent in a certain sense.  Instead, I think that the questions have a connection to the usefulness of the divide-and-conquer approach.
In complexity theory, a related notion to your notion of “recursive algorithms” with their “recursive runtime” is called “length-decreasing self-reductions” with their time complexity.
Some part of your questions (definitely the part referring to sublinear-time algorithms) depend on the choice of computational model.