How has the use and subsequent ban of DDT affected insect populations?
The most thorough study on the matter appears to be entitled "Anthropogenic impacts on mosquito populations in North America over the past century". Their broad conclusion is that various mosquito populations decreased in abundance and variety during heavy DDT usage years. However they have revived in the past few decades. The only species that seemed to have a permanent decline is Ae. Stimulans.  While this study was only on mosquitos, it may be representative of wider insects - most revived but a small fraction faced permanent damage.

There is a large amount of more associative evidence that observes a large decline in insect biomass and can only predict pesticides, such as DDT, as the culprit. The NYTimes had a good article on this.