What are the hairs/spikes on the edges of these leaves for?
I believe the hairs are known as trichomes. They can be uni- or multicellular. Depending on the plant and type of hair, they can serve one or more of several purposes. They may interfere with feeding by small and/or large herbivores, depending on how annoying to the palate they are. Some plants use hairs to keep frost away from the surface of the leaf. They can serve as nucleation points for water droplets in areas where the plant's primary source of water is fog. On the other hand, they can also help reduce transpiration in windy areas by breaking up the flow of air across the leaf. Finally, in hot/dry areas, they can reflect some sunlight to protect the leaf's tissues.

So, the function really depends on the type of plant and its habitat. I'm not a botanist, so I can't confirm whether or not your guess of Sempervivum is correct, but if it is, then the hairs would likely help reduce transpiration in hot, dry, windy areas, and possibly block some sunlight.