Zahl is any mathematical number -- you do arithmetic and so on with them.
Anzahl is an integer value (it's a type of Zahl)
Nummer is a number that names something -- house number, player #3 on a sports team.
For more, see the discussion at this link -- Hutschi and berndf give by far the best answers: http://forum.wordreference.com/showthread.php?t=1206849
I'm not sure what "read the numbers" means in this context. Does it mean: 1. The judge literally read the numbers. 2. The judge read the numbers out loud (e.g., the judge "read the sentence" means he said it out loud.) 3. "Read the numbers" as in "evaluated," i.e., "Yes, I received the report. I read the numbers and things don't look good."
I think DL isn't trying so much to make you understand and memorize specific sentences, but rather teach you how to construct a sentence on your own. If you understand the rules of how each part of the sentence works then you will be able to form your own.
That's why I try not to look too deeply into the meaning of the sentences but how the sentences are constructed.