In Adam’s little song of joy at Gen 1:23, is Adam naming Eve?

Yes, but as a woman, not as “Eve,” here (she is named later by Adam at Gen 3:20). In so doing, he is also, through the act of naming, demonstrating the sort of “dominion” he is said to have over animals. That is, he is naming a creation that God has brought before him, just as he had been naming animals God brought before him earlier. But this creature is received particularly joyously because she is a help meet, who came from his own flesh.