fungicide and fertilizer timing
Is this a home lawn? With a few exceptions (identification of a persistent disease), I do not recommend treating a home lawn with a fungicide. Having said that, you could use one any time after an insecticide treatment. That won't make a difference.
As far as applying fertilizer, it depends on the turfgrass species that you have. This link as a lot of information, but scroll towards the very bottom and there is a chart that lists the turf species and the pounds of nitrogen per 1000 square feet that should be applied annually. A typical range is 2-4 pounds. If you leave the grass clippings on the yard, that will be 1 pound over the course of the year.