Do bugs sleep?

-> Yes

"Bees have some of the most complex sleep states amongst insects. Decades of research show that insects do sleep, and that this resembles mammalian and avian sleep."

"Insects go through circadian rhythms of activity and passivity, but some do not seem to have a homeostatic sleep need. Insects do not seem to exhibit REM sleep. However, fruit flies appear to sleep, and systematic disturbance of that state leads to cognitive disabilities."