My son is in Kindy, and his assessment was done two days before school started. His teacher (as well as every other K student's) was already assigned, and it was his teacher that did the assessment.
It wasn't to place anybody in a "special" K class, it was just to let the teacher know where each student stands developmentally and academically coming into the classroom. Valuable information, IMO.
I wasn't allowed in the classroom during the assessment, but DS told me she asked him his colors and shapes. He was in there a half-hour though, so I suspect that she asked him a bit more than just that!
He didn't mention skipping, but now that it's brought up, I DO remember that being a skill MY kindergarten class had to do when I was 5. The only thing I remember about it though is that we had to show the teacher that we could skip across the room (which I thought was great fun!), and then the teacher teaching one certain boy how to skip one day. ("Step-hop! Step-hop!")