Yes, the yellowing of some lower leaves is normal and not a cause for concern. They are not a symptom of any disease or pest. Cut them off as low down as you can either at the soil line for where they attach to the main stem. Don't dig into the roots.
Without a photo of the bugs, it is hard to know how to treat them. Do you know the name of the insecticide you used?
Keep your Dieffenbachia in its plastic pot with the roots undisturbed. Keep it as close to an uncovered window as possible but just beyond the direct rays of the sun. Allow the top half-inch of soil to dry before watering it thoroughly.
