Pairs in common idioms/phrases

Solution 1:

One rationale for the pairings may just be that using them sounds better than a single word. They add a bit of rhythm to the sentence

Solution 2:

It is a figure of style known since Antiquity, called hendiadys, "one through two": two parallel nominal words are used to express a single idea, which would ordinarily be expressed by a "head" word and an attribute. There is no reason for this phenomenon but style.

Solution 3:

Off the top of my head, I would suggest two broad rationales for these idiomatic pairings: 1) contrast and 2) emphasis (based on the similarity between the words making up the pair).

Contrast

• checks and balances

• day and night/night and day

• do or die

• dos and don'ts

• give and take

• [come] hell or high water

• hill and dale

• high and low

• ins and outs

• open-and-shut

• pros and cons

• [come] rain or shine

• ups and downs

Emphasis

• airs and graces

• an arm and a leg

• be-all and end-all

• bells and whistles

• blood and guts/thunder

• [rain] cats and dogs

• cut and dried

• day and age

• dazed and desultory

• [between] the devil and the deep blue sea

• fire and brimstone

• flesh and blood

• high and mighty

• Ps and Qs

• plain and simple

• [between] a rock and a hard place

• simple and straightforward