Structure of -しい adjectives

Question

Are they all of the form [kanji]しい or can they also be of the form [kanji][kana]しい? ([kana]しい can also probably be found in the case of uncommon kanji.)

Answer 1

If you are asking the question from a purely orthographic aspect, the answer is written on the government's website.

Below are excerpts from the public notice issued in 1973 named 送り仮名の付け方. (Underlining in the original text is converted to italic, because there is no way to represent it due to SE's technical limitation.)

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語幹が「し」で終わる形容詞は,「し」から送る。
[(I-)adjectives whose 語幹 ends in し have okurigana from the し.]
〔例〕　著しい　惜しい　悔しい　恋しい　珍しい

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本則
活用語尾以外の部分に他の語を含む語は,含まれている語の送り仮名の付け方によって送る。（含まれている語を〔　〕の中に示す。）
[Words that contain other words in their non-活用語尾 portion follows the contained word on where to start okurigana.]
[...] 勇ましい〔勇む〕　輝かしい〔輝く〕　喜ばしい〔喜ぶ〕
[...] 頼もしい〔頼む〕
[...] 恐ろしい〔恐れる〕
[...] 憎らしい〔憎い〕　古めかしい〔古い〕
[...] 男らしい〔男)^[sic]

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（注意）
次の語は,それぞれ〔　〕の中に示す語を含むものとは考えず,通則1によるものとする。
[The following words are not deemed to contain each word in parentheses, and follows 通則1 (note: the first excerpt).]
明るい〔明ける〕　荒い〔荒れる〕　悔しい〔悔いる〕　恋しい〔恋う〕

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本則
複合の語（通則7を適用する語を除く。）の送り仮名は,その複合の語を書き表す漢字の,それぞれの音訓を用いた単独の語の送り仮名の付け方による。
[Okurigana of compounds follows (the result of) respective okurigana rules applied to each single constituent's kanji reading.]
[...] 聞き苦しい　[...]　待ち遠しい　　軽々しい　若々しい　女々しい

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許容
読み間違えるおそれのない場合は,次の(　)の中に示すように,送り仮名を省くことができる。
[Okurigana can be omitted when no confusion is expected, as shown in parentheses.]
[...] 聞き苦しい（聞苦しい）　待ち遠しい（待遠しい）

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I don't think I have been formally taught about these rules, but hardly see modern writing deviates from them anyway. Of course, the guideline assumes you have the understanding of Japanese vocabulary as a native speaker, but that should be made up for by trying dictionary.

One point is that a word that has etymological stem which is no longer valid in the modern language is generally not counted as rationale to push back the start point of okurigana. Same if the adjective have got its own kanji.

慌【あわ】ただしい (contains 慌てる)
甚【はなは】だしい (contains 甚だ)
夥【おびただ】しい (contains no living word)

Answer 2

Your question as written is more about spelling.

Looking more deeply at your intent, you could be asking whether ～しい adjectives are always derived from 体言 or uninflecting words, or whether they are also derived from 用言 or inflecting words.

～しい adjectives derived from 用言

• kuwashii, probably ultimately from ancient verb 構【く】う "to build up"
• urayamashii, from verb 羨【うらや】む "to envy, to be jealous of"
• misuborashii, from either 身【み】 or 見【み】 + 窄【すぼ】る "to be sunken in"
• konomashii from verb 好【この】む "to like, to prefer"
• yamashii from verb 病【や】む "to get sick"
• hoshii appears to be from verb 欲【ほ】る "to want, to desire"

～しい adjectives derived from 体言

There seem to be much fewer of these.

• otonashii from noun 大人【おとな】
• hisashii from noun 久【ひさ】
• hitoshii from number 一【ひと】
• nikunikushii from 憎【にく】憎【にく】, reduplication of the root niku of verb 憎【にく】む, adjective 憎【にく】い
• zūzūshii apparently from ずぶずぶ "dead drunkenly" from ずぶ "completely wet, completely soused"

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The above is by no means intended to be an exhaustive list, but rather just a sampling to show the broad patterns. In rough summary, ～しい adjectives have historically been formed from just about any part of speech: from verbs, nouns, adverbs. Verb derivations appear to be the most common.