You can eliminate paradoxes by declaring operations which lead to them "illegal"

Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They've eliminated self containing sets from the Naive Set Theory and Russell's paradox vanished. Hopefully.

Is i a number? If not, what is it?

It is not "what is it", it is "how does it behave". Whatever behaves as a number, is a number.

Does it have a sign?

If it behaves as it had a sign ... And it does. So yes, it has a sign!

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