The relative clause in which &c modifies torrent of radical thinking†: expressed as an independent clause this would be "the traditional image was displaced in this torrent of radical thinking".
† You might alternatively argue that it is only radical thinking which is modified; it doesn't make much difference.
The preposition phrase headed by by complements displaced, expressing the Agent of the passive verb: expressed in the active voice this would be "a structure of equal rights displaced the traditional image".
The relative clause introduced by where modifies a structure of equal rights: expressed as an independent clause this would be "in this structure individuals exercise autonomy".
As for the rest of your questions:
- The Subject is indeed The conflict between royalists and parliamentarians.
- English grammars which use the term 'Predicate' would say that everything from unleashed to the end of the sentence comprises the Predicate in this sentence; unleashed is the 'Verb' or 'Predicator'.
- Everything from a torrent to the end of the sentence is the Object. Its 'core' is the Noun Phrase a torrent of radical thinking, and what follows is a Relative Clause modifying the preceding Noun Phrase—which (as I said above) may be taken to be either the phrase headed by torrent or the phrase headed by thinking.
- In any case, there is no adverbial which modifies unleashed. In crucial areas of their lives may be taken as an adverbial modifying the clause headed by exercise. Traditional grammar might classify the Preposition Phrase headed by by as an adverbial modifying displaced; I would classify it as an agentive complement of displaced licensed by the passive construction.