I believe that if, for instance, you have two kids, you have an older and a younger - if you have three or more, you have an oldest, you have a youngest (and whatever is in between). I hear people constantly referring to one of their two kids as "oldest" or "youngest", and it hurts my soul - but I can't find proof on the Internetz that my soul is right. Is this a correct assumption?
I believe that if, for instance, you have two kids, you have an older and a younger
This part is pretty uncontroversial. You certainly do have a younger and an older (or elder). There are few who would dispute this.
if you have three or more, you have an oldest, you have a youngest
And again, uncontorversial.
The question is, can you also use the superlative for less than three.
Using it for the singular seems to generally be considered grammatical if most often redundant in the tautology involved: I don't have much reason for saying "I am the oldest adult male in my household" since I am the only adult male in my household, but it's not normally considered ungrammatical.
And likewise when there is an uncertain or varying number of people, that could potentially be two, the superlative is accepted. I could answer "yes" to "were you the oldest boy in your family?", with my having only one brother not being an issue here.
Those who object, object to the superlative being used in those places where it is clearly applied to one of a couple*.
And it's not surprising to find that this objection arose in the mid-18th Century and peaked in popularity in the 19th. It's very much of the fashion of the time that overlapping usages were frowned upon,† so since we can't use younger for the youngest of seven, there arose a corresponding rule of not using youngest for the younger of two.
But this is a "zombie rule"; a rule of English grammar that is proposed but was never actually followed. The Merriam-Webster Dictionary of English Usage quotes the 19th C grammarian Goold Brown from 1851 as saying it "...is not only unsupported by any reason in the nature of things, but is contradicted in practice by almost every man who affirms it." So we can see the rule was not accepted without any resistance.
And indeed great writers throughout the history if Modern English have ignored this "rule".
It is though reasonable as a preference. Sometimes using the superlative is less clunky than the comparative, especially when used along with superlatives because one is talking about different sets of which only some are couples. Often though the comparative is less clunky and genuinely more precise. I would favour it there myself, not because there's a rule insisting on it, but simply because that precision in communication can often—though not always—scan better. It's a matter of subjectively better rather than objectively more correct, but surely good English should soothe one's soul more than English that meets some set of technical constraints.
*There's also disagreement about whether a subset should be e.g. the "elder" group or the "eldest" group, though much of what I say above also covers that disagreement.
†Compare the nonsense written about not using less where one could use fewer, though the author of that "rule" expressed it quite clearly as a preference at the time, and would no doubt be horrified by the mangling some people do to follow it.
-est is for the superlative. It doesn't matter whether you are talking about ages of children (oldest, youngest), heights of buildings (tallest, smallest), or something else. You don't need any intermediaries, as the taller of two buildings is still the tallest, and the older of two children is still the oldest.
This usage is fine.
You may also find 'eldest' used in place of 'oldest.'