Sunrise/sunset data can be calculate yourself: https://gml.noaa.gov/grad/solcalc/calcdetails.html
As for the code/icons, there is indeed an 'icon' in the hourly forecast, for example:
From this url: https://api.weather.gov/gridpoints/PDT/31,35/forecast/hourly which returns an hourly forecast for someplace or other, there is this:
"icon": "https://api.weather.gov/icons/land/night/snow,20?size=small",