Part of the reason why the hummingbird is so amazing is because the speaker can't quite put into words what she just saw. A bird's flight-pattern leaves no concrete evidence that it took place.
A Resonance of Emerald (3)
This is where the speaker begins trying to describe what she just saw. Rather than just giving us a line like, "I remember it was green," the speaker describes what she saw and what effect it had on her at the very same time. The color is not just something seen, it resonates at a deeper level with the speaker.
A Rush of Cochineal (4)
Here the speaker highlights two reasons she is so awestruck—the bird's speed and its particular color. The word "Rush" also gives a sense of speed, as it's got an onomatopoeia thing going on. "Rush" is a great way to describe the sound of a hummingbird flitting through a flower bush.
And every Blossom on the Bush Adjusts its tumbled Head (5-6)
Now we get the effect the hummingbird has on the natural world. The blossoms get personified. They seem almost as amazed and dazed as the speaker is. Far out, flowers.
The mail from Tunis, probably, An easy Morning's Ride (7-8)
Even though the hummingbird is found in America, it gets the exotic treatment here, as the speaker imagines it coming all the way from Africa. It's as though, in her state of awe, the speaker is trying to pick the most amazing location she can think of to describe how easily this little bird might make a trip. And carrying a mail bag to boot!