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naive set theory (as far as I understand the history) led to some rather problemmatic paradoxes, which were remedied in stages by modifying the axiom system in use (proposed?)

Yes, but the axiom of regularity has nothing to do with it. Naive set theory is inconsistent because of the schema of unrestricted comprehension. The way the modifications avoided inconsistency is by not including the schema of unrestricted comprehension and instead using a weaker schema. The axiom of regularity plays no part in this.

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"proof by induction" (which apparently has the same strength as the axiom of infinity)

I dont' know what result you're referring to. There are lots of forms of inductive proof.

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"proof by induction" is only valid because we deem it so, we cannot derive it from the other properties of ZFC

I don't know what you have in mind here. ZFC does prove all kinds of inductive schemas. Even Z set theory, and even without the axiom of infinity, proves induction on natural numbers.

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we can form a heirarchy of cumulative hierarchies (of which ZFC forms only the first), and then once again (and again....).

I would put it this way, we can axiomatically extend the cumulative hierarchy.

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at some point we run out of names for these strange beasts.

We run out of names for things (in the language of set theory) as soon as we get merely to the power set of omega. That's quite soon in the hierarchy.

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forcing is used to show "equi-consistency" of ZFC+other stuff with ZFC.

Forcing was first used to show that ZFC+"not-axiom of choice" is consistent and that ZFC+"not-continuum hyphothesis" is consistent. Forcing since then has been used for many other consistency results. However, by the second incompleteness theorem, it is not possible to use forcing within mere ZFC to show the consistency of ZFC let alone the consistency of ZFC with any added axioms. [Note: I am not an expert about forcing, so I can't go very far in a discussion about it.]