Can every noun be a na-adjective?

Question

I know that na-adjectives are nouns which can also act like adjectives by affixing themselves to nouns, but can every noun do this simply by adding な or can only certain nouns have the property of being adjectival?

For 利口 can act as a noun:

利口 n. intelligence

but can also be an adjective:

利口 an. intelligent, smart, clever, etc.

Answer 1

Not every noun can be a na-adjective.

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A few examples of na-adjectives:

親切な人 -- kind person
安全な国 -- safe country
きれいな人 -- beautiful person
さわやかな風 -- refreshing breeze
ロマンチックな映画 -- romantic movie
健康的な食生活 -- healthy diet

Some nouns take の instead of な, and some categorize them as no-adjectives. You'd say:

病気の人 -- sick person
最後の晩餐 -- the last supper
年配の女性 -- elderly woman
初期の肺癌 -- early lung cancer
灰色の空 -- gray sky

... rather than 「病気な人」「最後な晩餐」「年配な女性」「初期な肺癌」「灰色な空」.

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In phrases like 「私の本」(my book) 「家の屋根」(roof of a house) 「日本の文化」(Japan's culture) 「空の色」(color of the sky), however, 「私の」「家の」「日本の」「空の」 wouldn't be considered as no-adjectives. These phrases would be considered as "noun¹ + case particle の + noun²" ("noun² + of + noun¹") rather than "no-adjective + noun" ("~~ish + noun").

And, you don't say 「私な本」「家な屋根」「日本な文化」「空な色」 etc. These nouns express もの・こと rather than 状態・感情, and cannot be na-adjectives.

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By the way...

I know that na-adjectives are nouns which can also act like adjectives...

Not all na-adjectives are nouns, eg きれい, さわやか, 静か, 明らか, ロマンチック, ～～的(健康的, 劇的, 科学的 etc.)

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As a side note:

You'd use "noun+な" before some words or phrases, even though the noun is not a na-adjective. In such cases the な is the 連体形 (attributive form) of the 断定の助動詞 (assertive auxiliary) or copula だ (source: 明鏡国語辞典). Some examples from the dictionary:

Words that mainly take "noun+な":
だけだ -- 「年が一つ下なだけだ」
だけに -- 「相手が子供なだけに対応が難しい」
ので -- 「まだ子供なので理解できない」
のに -- 「小学生なのによくやった」
ものか(終助詞) -- 「彼が犯人なものか」
もので -- 「まだ学生なもので」

Words that take either "noun+な" or "noun+の" depending on the meaning:
あまり・上・くせに・せい・ため・つもり・はず・ようだ・わけ・わり…

So you can say 「子供なので」「小学生なのに」「形容詞なわけではない」 etc. but it doesn't mean 子供, 小学生, 形容詞 are na-adjectives.

Answer 2

Technically yes, because you could use the form of a noun with な in saying things like すべての名詞が ナ形容詞な わけではない*, which is equivalent to …ナ形容詞であるわけでは….

However, that's not substantially an adjective. (For the rest, Tommy answered to your question, I think.)

*; This sentence is too unsettling to rephrase as 形容詞のわけではない while 形容詞のはずがない is fine.