When installing a single screw grip cap, what keeps it from spinning or moving? The Jerry Fisher grip cap has a couple of holes underneath. I suppose I could insert a dowel in the grip cap in the location of one of those holes. Is that necessary?

Exactly.
Those pin holes filled with stubby pins keep the thing from turning and allows the cap to be relocated accurately.
Stephen

I use bedding compound under the cap to keep it from turning. If it has holes on the bottom great, if it doesn't I drill some.