Regarding the first part of the question:
Have there ever been actual uses of q-calculus and quantum groups to computing or understanding solutions of Schrodinger equations, or functions actually arising from (physical) quantum mechanics, like states of the harmonic oscillator or some simple atomic hamiltonian like that of the hydrogen atom?
In my understanding, the answer is yes, there have been lots of applications in various different settings coming from quantum mechanical problems. As far as i know, the first applications of this kind were the algebraic descriptions of the solutions of the Schrodinger equations for (deformed) $q$-oscillators and $q$-rotators together with the computations of the corresponding energy spectrums, transition rates etc. Among the pioneering papers:
- The quantum group $SU_q(2)$ and a q-analogue of the boson operators, L C Biedenharn, J. of Phys. A: Math. and Gen., vol 22, 18,
- On q-analogues of the quantum harmonic oscillator and the quantum group $SU(2)_q$, A J Macfarlane, J. of Phys. A: Math. and Gen., vol 22, 21
Since then, there has been quite a lot of literature on similar topics. If we confine ourselves to low and intermediate energy QM (that is: mainly models of atomic and nuclear physics), an early overview can be found at:
- Quantum groups and their applications in nuclear physics, D.Bonatsos and C.Daskaloyannis, Prog. in Part. and Nucl. Phys., Vol 43, 1999, Pages 537-618
For a more complete list and further discussion on "actual uses of $q$-calculus and quantum groups ... ", maybe you will be interested in the answers (and the references included) to the following question:
Is there any published physics article where $q$-mathematics is applied?