In semiconductors with indirect band gap, when electron-hole pairs combine they usually just heat the material, instead of emitting light, which is why silicon, for instance, is not suitable for making LEDs.

While a direct band gap is desirable in LEDs, lasers and photodetectors, an indirect band gap is preferable in other applications where you do not want electrons and holes to recombine easily, e.g. in bipolar transistors or SCRs and in many kinds of diodes.

Cadmium zinc telluride is a II-VI semiconductor, not a III-V semiconductor, because Cd & Zn belong to the 2nd group, while Te belongs to the 6th group. (I find the habit of some modern authors of calling the group 2b as group 12 and the group 6a as group 16 extremely stupid, even if with the traditional approach it is debatable which should be group 2a and which should be group 2b, because for many properties Zn, Cd & Hg are more similar to Mg than Mg is similar to Ca, Sr & Ba. However this defect of the classic numbering is not solved, but it is made worse in the modern numbering.)

Both the III-V & the II-VI semiconductors, and also the few existing I-VII (made of Cu or Ag with halogens) and the few IV-IV semiconductors (e.g. silicon carbide) semiconductors, are compounds of chemical elements whose number of external electrons averages to 4, i.e. the same as in diamond, silicon or germanium, so they can form crystal structures of the same kind.

There are many other kinds of semiconductors, but those which have the cubic or hexagonal structures of diamond/lonsdaleite (more symmetric) or zinc sulfide (less symmetric) are much better understood than the other semiconductors and they are much more frequently used.

While a little strain can be beneficial in some cases, the large strain caused by the mismatches in crystal lattice cell size between various semiconductor layers that must be deposited one over the other in order to make some semiconductor device can cause great problems during manufacturing, by generating various defects that may make the process yield unacceptable.

Because of this, when researching new semiconductor materials a lot of effort is dedicated for finding compositions that can have matched lattice cell sizes.

Narrower FWHM means you will miss fewer energy peaks from isotopes.