I have the same issue. I toggled the following settings on and off many time.
Settings -> Lock Screen -> Always On Display -> Show Music Information
Settings -> Lock Screen -> Face Widgets -> Weather
Settings -> Lock Screen -> Face Widgets -> Show on Always On Display
During the switching, I even changed my clock style back to the default. For one moment, the song information showed up on the AOD screen, but when I changed my clock style back to my prefered one, it went away. Then I couldn't get it back, even after changing back to the defauly clock style.
Currently, all of the settings are Enabled, but no music information shows up on the AOD screen.
I've figured it out finally ! You gotta double tap on the clock on AOD, arrow will show up and you can click on it to look for the other information.
Speaking of music information, I'd seen it once appeared on main screen AOD without having to double click on the clock to search for it. It appear sometimes, and I am not too sure how it works.
Hope it helps ! Let me know if you've figured out the second one too.