In order for something to be recognized as a word, it needs to have meaning and be understood by more than 1 person. For brevity's sake, when I said "it's a word", I really meant "it's a word that's used and recognized by a large subset of people, large enough that lexicographers at dictionaries have taken notice".
"Irregardless" is not incorrect if enough people start using it that way. That's simply how languages work. You can have an opinion on specific words, but you cannot -assert- that using them is "incorrect". It's your opinion, not fact. Everyone also understands people when they use irregardless.
"These words are not self-contradictory, nor were they formed "as mistakes.""
Apron was originally "napron". It formed this spelling from people misunderstanding "a napron" as "an apron". Tornado should be tronado, because it comes from spanish tronada. Regardless (or irregardless? :)), the English language has many, many such words. My point is English already has so many words that are "mistakes", so arguing about "irregardless" doesn't make sense. No language used by humans is without illogical things.
"[Dictionaries] merely list words in alphabetical order along with their definitions"
Yes. They have criteria for when a -new- word becomes used enough that it enters the public lexicon, or an existing word gains a new usage. Like "literally" and "irregardless". You can't say it's wrong anymore when enough people are using it that way.