Is it significant that Eve is, just after being created, given to Adam as a wife (Gen 2:22-24)?

Yes, clearly. This clarifies that the sort of “help meet” that Adam needed was a wife. It also indicates another thing about the woman’s role: not merely a helper nor a mere companion, but one who is joined to the man, and who, as we will see soon enough, has a key role in multiplying the human race.