After Halo: CE, they retconned the "last Spartan" thing pretty quickly. A bunch of Spartan-IIs are still alive today, plus there are some Spartan-IIIs made prior to Reach that ended up surviving, PLUS now they are on the Spartan-IV program which uses consenting adults with military experience. The IVs are the ones that show up in the campaign.
The Covenant were on Requiem looking specifically for the Didact, actually. They worship him as the ultimate Forerunner god. They were fighting the Prometheans because they're not just gonna stand there while the Knights slaughter them, they're gonna fight back. Then when Didact shows up, he takes control of all the Prometheans and leads the remaining Covenant.