Does conjure minor elementals, with respect to the restriction on its CR, let the caster choose which elemental to summon? If no, is it determined randomly?
The DM Decides
Taking from Purple Monkey's answer to the linked question:
From the Sage Advice compendium:
Some spells of this sort specify that the spellcaster chooses the creature conjured. For example, Find Familiar gives the caster a list of animals to choose from. Other spells of this sort let the spellcaster choose from among several broad options. For example, Conjure Minor Elementals offers four options. The design intent for options like these is that the spellcaster chooses one of them, and then the DM decides what creatures appear that fit the chosen option.
Conjure minor elementals is explicitly listed as an example of the sort of summoning spell where the DM is the one to decide which creatures appear.
Technically, the DM doesn't even have to let you have any say over what element the creature is related to, since it says (PHB, p. 226):
You summon elementals that appear in unoccupied spaces that you can see within range.
as opposed to conjure elemental (PHB, p. 225), which says:
You call forth an elemental servant. Choose an area of air, earth, fire, or water that fills a 10-foot cube within range.
and therefore the element of the creature is determined by where you cast the spell, for conjure elemental. Since conjure minor elemental just lets you summon it anywhere, the DM gets to decide the element type too.