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The following pairs of inflections got me thinking a little bit.
Both of these suggest that there is perhaps some historical connection between the く sound and い sound, either phonologically or semantically. Is there any reason why い and く seem to like to "hang out" with each other like this? Maybe these two classes of words diverged from the same class of words somehow?
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) only intended to discuss the phonology, whereas the portion of your question here is driving towards something deeper.
In the modern language, these adjectives have three main endings: ～い for use in the predicate (sentence-ending) or attributive (directly modifying a noun) roles, ～く for adverbial, and ～かった for past tense. Notably, ～かった is actually a contraction of adverbial ～く + あった, the past tense for classical copula ("is" verb) あり, modern ある. So we're left with two endings, just ～い and ～く.
However, these ～い adjectives used to have three endings.
Some time during the Kamakura period (1185–1333) the medial (mid-word) //-k-// sound started disappearing in certain phonological contexts. This happened for both the ～き and ～く adjective forms, and for the ～き on the end of verb stems (the part before the ～ます) when followed by the conjunctive ～て ending. This produced things like 書【か】きて → 書【か】いて, 強【つよ】き → 強【つよ】い, and 早【はや】く → 早【はや】う. From what I've read and can recall, it seems this shift may have started in the Kansai region, and slowly gained adoption elsewhere. This produced the modern standard Tokyo-dialect adjective ending ～い and the well-known slightly-irregular ～て forms for verbs.
This loss of //-k-// plus a flattening of vowels also produced terms like おはよう and おめでとう, from honorific お～ + the no-"k" adverbial forms 早【はや】う and 愛【め】でたう.
For reasons I don't yet understand, however, the loss of medial //-k-// for adverbial forms did not gain full acceptance, and thus we still have that //-k-// in the adverbial ～く ending for adjectives in standard Tokyo-based Japanese. I believe the no-"k" adverbial is still in common use in Kansai, but I must rely on others to confirm, as I have never spent time living in that part of Japan.
Although the shift here is about a loss of medial //-k-//, in Japanese, this is described as a shift towards the vowel value. So the き → い change is called イ音便【おんびん】 (literally "i sound passing-along", or "i euphony"), while the く → う change is called ウ音便【おんびん】. Read more about Japanese sound shifts at the Japanese Wikipedia and the English Wikipedia.
Ultimately, the only connection appears to be the phonology. I have never read anything suggesting that ～い adjectives as a class have anything to do derivationally with verbs ending in ～く.
Please comment if the above does not address your question.
Yes, for example consider the beginning of 枕草子【まくらのそうし】:
美【うつく】しきもの。瓜【うり】に描【か】きたる稚児【ちご】の顔【かお】。
If you rewrite the adjective and verb forms to their modern form:
美しいもの。瓜に描いてある稚児の顔。
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 see the same shift き→い.
