How can insects survive without an adaptive immune system?
There is not really a definitive explanation for why, although it's important to note that many mammalian pathogens are not adapted to insects and vice-versa. Insects need to survive insect pathogens, and they have a number of defenses for this purpose. Animals and their pathogens co-evolve and exert selection pressures on each other.

Killing the host (or killing the host quickly) is often not in the pathogen's best interest. Pathogens evolve towards their own successful reproduction as much as animals do. Success may ultimately mean evolving towards a less pathogenic state that improves a pathogen's chances of being transmitted.

As animals develop new defenses, they gain an advantage (relative to others of their own species) in surviving various pathogens. Pathogens evolve to escape these defenses, and a new equilibrium is established. An animal lacking this defenses is likely to be more susceptible to dying of infection, as the new equilibrium is dependent on them having these protections. 

Here is an interesting paper on this view: http://hedricklab.ucsd.edu/PDF/HEDRICKImmunity113.pdf