The Elites, Jackals, Grunts, and Hunters we see are apart of a radical faction. They don't necessarily represent The Covenant anymore but they do view themselves as The Covenant.
I assume after the truce was made between The Covenant and Humans, The Covenant packed up and left Earth. It's never mentioned but it is implied.
As for the UNSC Infinity, the first live-action trailer technically explained the purpose behind The Infinity. From what I could tell, the ship was created shortly after Human-Covenant War.