What are the applications of edge states in 1D topological systems? In 2D, we get robust conducting edges. In 3D, we get robust conducting surfaces. These are interesting because we can possibly utilise this robustness for protected electron transport (or light transport in photonic systems). 
What are the applications of 1D edge states (when the wave function is localised at the edge)? 
I believe Kitaev's 1D model has quantum computing applications (how does this use the edge state?), could other 1D systems be used in a similar manner? Are there any other applications of 1D edge states?
 A: As you mentioned, one notable application of 1D edge states is for quantum computation, so I'll provide some more details on this.
In traditional routes of building quantum computing platforms, the qubit states $|0\rangle$ and $|1\rangle$ are encoded in two different energy levels of a physical system (e.g. atomic levels), and operations/gates are performed by driving transitions between these levels (e.g. with laser pulses). An important challenge for these approaches is that the energy splitting between $|0\rangle$ and $|1\rangle$ states can cause one state to decay into another (or even to states that are not part of the qubit two-level system), especially during gate manipulations.
In contrast, proposals have been made to encode qubits "topologically" in 1D edge states (a good reference is https://www.nature.com/articles/npjqi20151.pdf). In this case, two ends of a 1D nanowire encode a single qubit. The two energy levels are degenerate and separated from other states by a finite energy gap, making the system robust against weak local perturbations. Furthermore, gates can (in principle) be implemented by exchanging one of the edge modes around another. The operation applied will then depend only on the number of times one quasiparticle is moved around the other and not the implementation details, so it is also "topologically protected".
