Best sources / methods to find additional cloud information for current weather conditions?
The Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite provides new insight into the role that clouds and atmospheric aerosols (airborne particles) play in regulating Earth's weather, climate, and air quality.

CALIPSO combines an active lidar instrument with passive infrared and visible imagers to probe the vertical structure and properties of thin clouds and aerosols over the globe.

The data from the satellite can be found here (https://eosweb.larc.nasa.gov/project/calipso/calipso_table), from what I see, the latest dataset is about a week old. While this won't give you the cloud type, it will surely give you the cloud heights, you will have to apply some knowledge though to differentiate between clouds and aerosols. 

For the cloud type, I suggest you can look at visible satellite data or infrared satellite data to look at clouds and judge the cloud type.

This is the e-wall from Penn State, 
http://mp1.met.psu.edu/~fxg1/ewall.html
If you look at the left panel about mid-page, you see the heading 'Satellite' and you can look at the visible and infrared spectrum and also a water vapor satellite image.

This approach seems very raw, but this is what I could remember off the top of my head. I don't know of a page that provides real time data about cloud height and cloud type.