Pre rain is fine for lime as long as it's not a downpour, otherwise, you will just raise the Ph of the storm drains. The same goes for crabgrass preventer. That has to be watered into the soil to do it's job of sterilizing seeds before they sprout..
You do not want any rain after putting down granular weed killer. It has to be on the leaf for it to work. Best practice is to put down with at least three rainless days in the forecast, on a just watered or dew covered lawn to get the granules to stick good.
Liquid weed killer is different. Most have a wetting agent that makes it stick to the plants and stay, it is rain fast in 24 hours.