The verb picar is definitely the most common (in my experience, in Spain) for insect (and related) bites. However, morder is also used. For example, Wikipedia shows 211,000 hits for "picadura de araña" vs. 147,000 for "mordedura de araña".
Like in English, picar is more associated to a stinger and morder to a mandible or biting device. Notice that mosquitoes pican, because although they do it for feeding purposes and using their mouth organs, these are needle-shaped and hence picar.
BTW, and slightly off-topic, scorpions and spiders are not insects but arachnids. ;-)
With regard to other animals, most people would commonly use picar for snakes, but it's more proper to use morder in this case. Fish with stingers pican, of course.