1 - To what extent is DFT is useful if one wants to understand questions, such as topological quantum computing? Is DFT a good starting point if one is interested in studying aspects of condensed matter, such as quantum spin liquids, topological quantum computation, etc. ? or is it better to study these systems using tight-binding lattice models without going through the DFT phase?
It largely depends on what specific questions you're interested in, and where on the abstract-to-concrete scale you want to work / feel at home. Part of the beauty of condensed matter physics is its broad span in topics, tools, and energy and length scales. In a lot of cases DFT or other quantum chemistry methods can be sufficient, but in other cases it's not a very natural starting point. For a graduate student starting out, I would recommend finding a problem that interests you deeply, and then learning methods appropriate to that problem. Maybe those are pen-and-paper analytical methods, maybe they are DFT methods, maybe something else. There are a lot of possible paths.
For example, if you want to understand the mathematics and principles of quantum spin liquids, you're usually better off focusing on effective low-energy theories, e.g. spin models and lattice gauge theories. There are specific methods well-suited to analyzing such systems. Similarly, the language of topological quantum computing is often framed in its own set of effective low-energy models. At this end of the scale, we are often interested in new emergent phases and quasiparticles that tend to be hard to capture or notice in more realistic models. However, these low-energy models can often be obtained as a limit of a more concrete model.
For example, the Kitaev chain was originally written down as a toy model for a system that might be used as a topological qubit. It was subsequently realized that this model emerges at low energy in e.g. spin-orbit-coupled semiconducting nanowires when subject to a magnetic field and proximate to a superconductor. But note what the ingredients of that recipe are: nanowires and superconductors. To go even more concrete, we need to think of actual materials which may be used to realize these components. Similarly, Kitaev's honeycomb model hosting a quantum spin liquid was first considered a toy model. It was then argued it could occur in systems of certain crystal structure and electron configuration. DFT is very useful in finding materials realizing those conditions, and thus candidate materials to realize the Kitaev spin liquid. But it'd be very difficult to start with a DFT calculation taking into account complexities of real materials and try to infer spin liquid behavior directly.
2- How important will DFT be in the near future? are people looking for alternatives?
DFT is not going away anytime soon. It's still fast compared to similar methods, useful, and continuously being improved in different directions. But it has its limitations. These weaknesses do motivate a lot of work on e.g. so-called post-DFT methods. Overall, the space of possible materials and systems we want to model is vast, and realistically this requires a variety of approaches.