The double negation of i-adjectives not only exists, but it is quite commonly used among us native speakers when expressing opinions indirectly.
Take 「おいしい」 ("tasty") for example, by far the most common double-negative form would be:
"(the food) is okay/passable if not great"
That sounds fairly indirect, doesn't it? The ...
Yes, the conjugation rule is consistent with all the い-adjectives.
There are a few notable exceptional cases where they can also syntactically be nouns:
adjectival: 近くのX - the nearby X
noun: Xの近く - the vicinity of X
there is perhaps some historical connection between the く sound and い sound, either phonologically or semantically.
I think the answer from blutorange addresses this.
Maybe these two classes of words [〜い adjectives and 〜く verbs] diverged from the same class of words somehow?
I'll disagree with blutorange about this part, as his answer is (I believe) ...
Yes, for example consider the beginning of 枕草子【まくらのそうし】:
If you rewrite the adjective and verb forms to their modern form:
Observe that an adjective such as 美しい was originally 美しき (before a noun, 美し at the end of a sentence), with a shift of き→い, see イ音便. 咲いて was originally 咲きて, where you can ...
I'm the one who wrote the majority of that article (some ~10 years ago!), so I apologize if that section was a bit confusing when comparing the Kagoshima forms to their standard Japanese counterparts.
As @mamster pointed out, for some of the entries in that table, the root word used in Kagoshima is completely different from that used in standard Japanese. ...