Quantum and organic chemistry of vitamin D synthesis
These are pericyclic (i.e. concerted) reactions. They do not need particularly high temperatures, but indeed they can be catalysed by either the right photons or temperature. They are not "reactions" per se, rather just intramolecular rearrangements of electrons over a "chain" of connected atoms.

The first reaction is a photolytically catalysed electrocyclic reaction that results in the opening of the 6-membered ring, they are very common.

The second reaction is a thermally catalysed [1,7] sigmatropic migration of the H. The rate is dependent on the nature of the nucleus that moves, in this case H (D would be slower). Quantum tunneling will indeed enhance the rate of its non-quantum counterpart, but it is not the driving force of the step as you seem to wonder.

As for the reason why these reactions happen in the first place, it's because there is favourable overlap in the transition states between the orbitals of the atoms at the two external ends of the "chain" of atoms interested in each rearrangement.

As you can see below, the HOMO in each case permits favourable overlap (phase-wise and sterics-wise) between the ends of the "chains" in both cases. I couldn't quickly find a picture of a photolytic ring opening but in that case the actual HOMO is a SOMO: one electron from the former HOMO has been promoted (by the photon) to the former LUMO. This will change the overall symmetry properties of the atom chains so that a [1,7] H shift is allowed under photolytic conditions.

Electrocyclic reaction:


Hydride shift:


Reference: https://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/pericycl.htm

Suggested reading: "Pericyclic Reactions" by Ian Fleming (OUP, 1998).