Do they light up the face of mount Rushmore after dark?

Yes, From 9:30 to 12 o'clock

Yes.they do

Nowadays i don't think so they do because there's construction going on

Yes they do, there's a lighting ceremony every evening during the spring & summer months. More information and times are available at https://www.nps.gov/moru/planyourvisit/evening-lighting-ceremony.htm