Word-Usage – ‘Mine Alone’ or ‘Mine and Mine Alone’

You are mistaken in supposing this to be a recent innovation: to the best of my recollection it arose in the late 1950s. At that time it was supposed to be characteristic of the ‘beat’ community, but it entered youthspeak almost immediately and has been current among the young ever since, at least in the United States (I cannot speak to other dialects). As teenagers grow older and are called upon to address more diverse audiences more formally, they generally learn alternative expressions (as it were, so to speak, I think, in my opinion) culled from severer registers; but like never entirely disappears from speech with close friends.

It has a useful and indeed essential function in speech as a discourse marker. When it first arose it was employed primarily as an anticipatory emphatic, to signal that what follows will be of particular importance—Maynard G. Krebs’ favorite expression was “Like, wow!”—but by the end of the 60s it had acquired additional anticipatory uses: notably 1) as a signal that a subsequent pause will not mark the end of an utterance but further processing and 2) as a lexical ‘quotation mark’.

There is thus nothing to deprecate in the usage except snobbish preference for more prestigious markers. If the usage annoys you, I suggest you seek a more pretentious class of interlocutors.

"Core" is a common word, most often used to refer to either the center of something, or the main (most integral) part of something. Consider "strengthen your core", "core curriculum", "at the core of the issue". The words existed long before computers, so I would suggest looking at their usage in that context as descriptive, rather than prescriptive. We didn't start using the words because of what they meant in the context of a computer, we named the parts of the computer based on someone's interpretation of how we already used the words.

"Nucleus" is used less often, though anyone with an elementary knowledge of the atom would understand its meaning. The nucleus is at the core of the atom, so you're correct that the two words have been extrapolated to have similar meaning in certain contexts. I would say that nucleus is more often used to describe the actual center of something, rather than "the most important part". There is also another popular usage of nucleus, where it has a definition similar to that of the word nexus (defined as "a relationship or connection between people or things"). In these contexts, it is often interchanged with the word cause:

cause, noun

a person or thing that acts, happens, or exists in such a way that some specific thing happens as a result; the producer of an effect: You have been the cause of much anxiety. What was the cause of the accident?

In casual conversation, I wouldn't expect to hear the word nucleus very often. It would be more likely to come up in the second use case, likely while trying to explain a point.

The most common understanding of the word kernel in daily conversation is:

the softer, usually edible part contained in the shell of a nut or the stone of a fruit.

In context, your audience would likely pick up what you meant if you used kernel in the core/nucleus sense. But I don't think I've ever heard it used that way myself, unless actually referring to a scientific kernel. This doesn't mean you can't say it, just that it isn't common.

Most of the time, core is going to be your best option. It has the broadest definition, and is used in many areas (core inflation, etc., as you mentioned). I don't think you need to use your other day-to-day suggestions instead, core is a perfectly understandable and common way to express what you're getting at. You might consider alternatives such as your suggested "main" or "central" if you feel there might be ambiguity regarding the definition of core you intend to use ("main/most important" or "central concept").