If 好き is a noun, then why does it take the な copula in things like 好きなこと? I thought that copula was reserved for adjectives, and I thought nouns took on the の copula.
I think 好き was originally a noun derived from 好く, but then it came to be used as a na-adjective as well. In fact, most na-adjectives derive from nouns (and some people consider them to still be nouns).
The following quote is from Origins of the Verbalizer Affixes in the Japonic Languages by Tyler Lau:
The Uehara paper cited is A Diachronic Perspective on Prototypicality: The Case of Nominal Adjectives in Japanese (Satoshi Uehara, 2003).
So it's not surprising that 好き looks like a (deverbal) noun. But if you check a dictionary, you'll see that it's marked not only 名 (short for 名詞 "noun"), but also 形動 (short for 形容動詞, which is the part of speech that corresponds to na-adjective).
So from the perspective of the modern language, it uses な (or だ, etc.) because it's a na-adjective.
The な suffix is reserved entirely for nouns. な adjectives behave exactly like 名詞 in my experience, aside from using な when modifying another noun. 好き in English is approximately equal to the noun 'desirability'. This, however, is an inconvenient explanation to young learners, so it is common to see the 形動 explained as 'な adjectives'. The problem is, 形動 do not interact with other Japanese words the way that Japanese adjectives (い adjectives) do.
Fundamentally speaking, there are three main classes by which words in Japanese interact. The first, 'nouns', broadly includes 名詞 and 形容動詞 (な adjectives). The second, 'verbs' includes 動詞 and 形容詞 (い adjectives). The third, 'adverbs' roughly includes any adjectives, particles, postpositions, 形容動詞 with the と suffix, etc.
The only apparent difference between both of the two (aside from English equivalent) is that so called な adjectives use な in place of the possessive の.
With verbs, the differences are much more apparent. い adjectives cannot use the を particle, but may take the suffix です for politeness. Besides that, they may both be conjugated, and may serve as subordinate clauses to other nouns. You may also end sentences with any verb.
I won't get into 'adverbs' here as the class is very general and honestly, quite irrelevant to your question. Suffice to say, there is a humongous variety of 'adverbs' as I have basically used this class as a catch all for semantic modifiers of every description.
な adjectives are Japanese nouns which take な when modifying another noun. い adjective are Japanese verbs with unique conjugation rules. I'm sorry to say, but because these words translate into English adjectives, numerous beginner's materials make the mistake of naming them both adjectives. The honest truth is that 形容動詞 are a special subclass of nouns. Contrarily 形容詞, the い adjectives, are actually a lot more similar to verbs.
According to lesson 40 in "Making Sense of Japanese Grammar: A Clear Guide Through Common Problems" by Zeljko Cipris and Shoko Hamano, the な suffix belongs to a small set of nouns that due to their abstract nature, are given a special suffix. For example, you can't deny that a student belongs to a university (大学の学生), but you can argue about what 'healthy' is (元気な人). General speaking, only the most subjective of nouns use な. (sickness is easier to identify than health, explaining the contrast between 元気な～ and 病気の～.
This link may be of interest: http://www.yano.bc.ca/vansin/vansinpo072.htm