1. Those are completely different sets of words, some of which look more similar than others, some of which have languages related to each other (and others don't), and some are explained by other reasons.

2. Randomly picking words that "look similar" is not a good scientific approach. If you want to show any meaningful relationship, then you need a lot of systematic data showing patterns.

As for these specific sets of words, here are the answers:

아빠(korean) - father - بابا/اب(Arabic) - faðir(icelandic)- papa
엄마(korean) - mother(English) - ام/ماما(Arabic) - mama - móðir(icelandic)

These are similar because these are the first words babies say. The "m" and "b/p" sounds are the first that babies produce, and they first say "mama" then "baba/papa". This is found in dozens or hundreds of languages around the world. These words are more like onomatopoeia than "mother" and "father", which are better for comparing language relationships.
https://en.wikipedia.org/wiki/Mama_and_papa
https://www.quora.com/Why-do-papa-and-mama-sound-similar-in-different-languages-while-son-and-daughter-dont/answer/Daniel-Ross-71

아니오(Korean) - no(English) - non(French) - нет(Russian), nej(Danish)

English, French, Russian and Danish have a shared ancestor (Proto-Indo-European), and this word goes back to an older negation word in that language. Korean is a coincidence.

Shi(Chinese)/Si(Spanish), da(russian etc), yes(english)/já(icelandic)

These words do not look similar. The only similarity is that they have a consonant-vowel (CV) pattern, which is easily explained because they are frequent words, and therefore likely to develop short forms over time. There is nothing more to explain here.

I'd suggest reading an introductory textbook about historical linguistics and linguistic reconstruction. I've written something about that here:
https://www.quora.com/How-can-we-learn-the-pronunciation-and-sound-changes-of-different-languages/answer/Daniel-Ross-71

No.

"Sandhi... is a cover term for a wide variety of sound changes that occur at morpheme or word boundaries."
https://en.wikipedia.org/wiki/Sandhi

Those are just contractions. More technically, they're part of a (diachronic) cliticization process where originally independent words become attached to others as dependent morphemes, like affixes, but still somewhere between morphology and syntax.

A few of those have some kinds of sound changes ("won't" for example) that you could call "sandhi" but mostly that would be missing the overall point.

I discover some word in many language  is similar
Why are this words in many language similar?

아빠(korean) - father - بابا/اب(Arabic) - faðir(icelandic)- papa
엄마(korean) - mother(English) - ام/ماما(Arabic) - mama - móðir(icelandic)
아니오(Korean) - no(English) - non(French) - нет(Russian), nej(Danish)
예(korean) - yes(english) - já(icelandic) - ja(Danish etc)

Sure. But how can you "prove" a construction? Regardless, having a specific construction wouldn't mean that in the general case this isn't just a 'citation'/mentioning form.

Still, I don't think either of us has a direct counterargument to the other's analysis, or any way to (potentially) falsify either one. So we can just leave both as plausible explanations.

This isn't a specific grammaticalized construction. It's possible in any context where citing/mentioning a complex NP is pragmatically appropriate instead of referring to individual entities. Literally any sentence/construction you can imagine would permit this, although of course rarely does English mark grammatical number so that it would be contrastive. Basically it would only ever appear with subject NPs. I don't know whether other languages behave similarly.

If you were organizing a photo shoot for some kind of advertisement and trying to decide who to put in the photo with the product, and you wanted to explain to someone about who to put in the scene, would this be appropriate in Italian?
"Il bambino e la bambina è perfetto."
If not, Italian is different from English. If so, wouldn't the same sort of mentioning-rather-than-referring apply?

Actually going back to something you said earlier, we could think of this as a sort of grammatical metonymy: we refer to the singular concept of a collection of entities in the singular because that is how they are pragmatically relevant.