Is there a way to program an ICE (in case of emergency) number that is accessible without unlocking the phone? Not 911, but a family member in my contacts?

Not directly, but you can create a lock screen wallpaper that displays instructions and a phone number. And you can actually dial any number from Emergency Call, not just 911.

I guess not. It used to work. Some release changed the way it works.

you can now set up an Emergency Contact in 'Medical ID' in the Healh app. When someone goes to the lock screen and switched to the Emergency call, they can click on the link Medical ID and then contact your emergency person. It also lists your age, allergies, and illnesses if you need it to.