It's purdy! Some desert plants grow spikes and thorns to hold on to their growth and moisture, and others hold the water in their tissues, mix it with foul-tasting sap, and then coat themselves with a waxy substance to prevent predation. In the desert, if you have anything work keeping you have to fight to keep it.

I'm not real sure what this specific plant is, but if the side shoots you refer to are coming up from below ground, they're buds or pups. A common way many succulents reproduce (such as aloe). If they're just branching off from the main stem you can clip them off without worrying about it too much. There's always a chance of disease whenever you have a plant injury like that, but should be a low risk.