They are not baby spiders, but you are not too far off target. They are Spider Mites. Spider mites drink the sap of plants and in great enough numbers they can cause severe damage. The webbing is also from them, they produce it in an attempt to protect themselves from predators and to shield themselves from unfavorable climate conditions. Spider mites are not insects, they are actually more closely related to spiders and scorpions (all three are Arachnids).
Your insecticide, imidacloprid, is specifically targeted to insects and has no direct impact on mites (Source). Many insecticides can actually make a spider mite infestation worse, because many of the spider mites natural predators are insects. If you spray to kill the insects you may remove beneficial insects that are trying to eat your mite infestation. If that weren't bad enough some insecticides, like Carabyl actually encourage mites to reproduce faster.
Spider mites strongly favor dry, hot conditions. Also they tend to show up the worst on plants that are already suffering from drought stress, which may or may not be relevant to your current situation. Neem oil, as well as several other kinds of insecticidal oils seem to be your best option. Make sure you get good coverage, as it only effects the mites it hits.