Does the need for renormalization in QFT vanish once you use a more fundamental theory (e.g., string theory)? It is often explained that renormalization arises in QFT because QFT is a low-energy effective theory that needs to be replaced by a more fundamental theory at higher energies/smaller distances. While we don't have a more fundamental theory that's accepted by everyone, candidates do exist. Can string theory for example handle the same calculations as in QFT but without renormalization? If I have a Feynman diagram that diverges in QFT and I replace the point particles by strings, will I be able to now calculate the diagram without issue?
 A: What I think one needs to internalize conceptually is that the program of renormalization is always favourable (and almost always required) in physical theories, be they fundamental or effective phenomenological ones (including condensed matter field theories), be there infinities or not.  
I think the last point is by far the most important. Yes, renormalization did formally arise as a method of handling divergent loop integrals by declaring only physically measurable observables be finite and well-behaved. This is well justified for a physical theory, as can never actually measure the action or lagrangian itself, just the scattering amplitudes which in turn provide us with the Green's functions or Correlation functions of the theory.  
For a moment switching to condensed matter systems, which do have a finite UV cutoff, there are no divergences in the perturbative formulation of the field theory, as we always have some lattice (or equivalent discrete structure) at the shortest length scales. Even with the absence of infinities, we do renormalize such theories, the essential reason being that renormalization is a procedure by which one can decouple the low-energy physics (the IR behaviour) from the high-energy UV one (that takes place on the scale of, say, the lattice spacing). It is this removal of sensitive dependence on the microscopic details that underlies the idea of renormalization.
So, even if a theory is UV complete, which is something one would require of a fundamental field theory, we would want to renormalize the couplings and calculate their flow so that your everyday coffee may not spill because a particle collider discovered new interactions at the Planck scale!
A: You seem to be confusing regularization with renormalization.
Regularization is the process of removing (or, more properly, parameterizing) infinities in loop integrals. Often in elementary texts a "cutoff" representing an energy scale above which the theory is assumed to be invalid is discussed, and counterterms are added to the Lagrangian in order to make loop integrals finite.
This introduces ambiguities in the definition of the theory parameters, like masses or coupling constants. Enter renormalization, which is the process of carefully defining what it means to measure a parameter so that we can properly define a Lagrangian that gives the correct results for measured quantities.
While often discussed at the same time, renormalization and regularization are completely separate and distinct procedures. Consider a loop integral for a scalar field in 2 dimensions. Once Wick-rotated to Euclidean space it would look something like this: 
$$I(p) = \int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2 + m^2}\frac{1}{(p+k)^2 + m^2}$$
For large $k$, the integrand goes like $\frac{1}{k^4}$ so there's no divergence. However, one loop effects such as this would still screen charges or modify masses and you would need to perform renormalization in order to connect theory and experiment. Renormalization really is an established part of physics, with observable consequences. No future developments will ever get rid of it. The couplings really do run, the masses really do get radiative corrections, and so on. String theory is no different.
The requirement for a fundamental physical theory is not that it doesn't require renormalization -- an empirical and logical impossibility -- but that it be UV complete. This is the requirement that the theory is well defined up to arbitrarily high energy scales and that it is predictive (typically taken to mean "renormalizable" but fashions change). String theory is known to be UV complete. Asymptotically free field theories are also known to be UV complete.
Also see What is the definition of a "UV-complete" theory?, the Schwinger Model and Does string theory provide a physical regulator for Standard Model divergencies?.
