Your main question
How does 形容詞+かる work in Classical Japanese?
In basic terms, the conjugation charts in the Japanese Wikipedia article at https://ja.wikipedia.org/wiki/形容詞#古典日本語の形容詞の活用 are quite useful. Reproduced here:
The so-called ク活用 or "-ku conjugation" is for adjectives that (in modern Japanese) end in just ～い, such as よい or さむい. Meanwhile, the シク活用 or "-shiku conjugation" is for adjectives that (in modern Japanese) end in ～しい, such as あたらしい or さびしい.
Fused with aru
Fused with aru
Irrealis (hasn't happened yet)
Continuative (-masu stem)
Realis (as if it's happened)
I remember reading previously that かる could not be used attributively...
I'm not sure where that might have come from. As you can see in the table above, the ～かる or ～しかる form is indeed the 連体形【れんたいけい】 or "attributive form", and this is used attributively in historical texts. Granted, the "basic" attributive form ending in ～き or ～しき appears to be more common for regular usage of an adjective to modify a noun, but we can find instances of ～かる directly modifying a noun, as in this text from 1949:
For what else could a 連体形 have been used in the first place?
Jimmy Yang's comment on your question links through to a page that does a good job of providing a nutshell explanation of what the 連体形【れんたいけい】 was used for. There is also more here at Kotobank and here at the Japanese Wikipedia. (Sadly, the English Wikipedia article section here is a bit of a mess, so please only read that with a grain of salt.)
Briefly, in English, the 連体形【れんたいけい】 or "attributive form" of either a verb or adjective could be used to:
- modify a noun or noun phrase, basically just being an adjective.
- act as a noun itself -- this may have evolved by an elision (omission) of an abstract noun like こと ("fact, thing"), which the attributive may have originally modified.
- serve as an attributive, but coming after the thing it modifies, in a special kind of Old Japanese grammatical construction called 係【かか】り結【むす】び.
Was [the -karu form of the rentaikei] simply an alternate form of ～き in its zero-nominalisation?
I'm not familiar with the term "zero-nominalisation", so I cannot answer you here.
Was [the -karu form of the rentaikei] placed where neither an adjectival 連体形 nor the 終止形 would fit, like in ～と and ～可し?
I'm not aware of any cases where と requires the ～かる form of an adjective.
Note that the -karu ending is a fusion of regular adverbial ending ～く + the attributive form of Old and Classical Japanese copula ("to be" verb) あり. I wonder if the requirement of this form for certain 助動詞【じょどうし】, such as べし or らむ, might be because the 助動詞 itself requires a verb -- which is provided with the ある in ～かる, but not in the "basic" attributive ～き ending.
Was [the -karu form of the rentaikei] utilised as an emphasised 終止形, similarly to forms of the 連体形 in other constructions?
I have no idea what other "forms of the 連体形 in other constructions" you are referring to, so again I cannot answer you here.