My understanding is that perturbative QFT can essentially be described as a weighted sum over 1-D topologies (ie Feynman graphs), and String theory is essentially the generalization to a sum over 2-D topologies. Why do we stop here? Is there a name for the theory defined as the sum over 3-D, or N-D topologies?
There is. They're all included in the name "string theory".
While it originally began as a theory of 1-dimensional strings, today it describe a quantum theory of many other $p$-dimensional (D$p$-branes, membranes, etc.). We just didn't bother finding another name for the theory. The key difference is that while a fundamental string is perturbative, the other objects are not (since they are very heavy)