I graduated from school 523.
How should I read the school number? Should I say
1) school five two three
2) school five twenty-three
3) school five hundred and twenty-three
... or some other way?
If you have a bad connection, "five two three". That's a somewhat general rule for any number if it might be difficult to understand: say the individual digits.
Generally, numbers above 100 have some digits said individually, or grouped in pairs. So one would usually say "I went to PS [public school] five twenty-three." Or, "...fifty-five twenty-three" (5523). "...five fifty-five twenty-three" (55523) and so on. Avoid saying the word "hundred" or "thousand", if possible. [EDIT] One exception to that: for numbers where there are no more than 2 leading digits followed by zeros, people frequently do say "hundred" or "thousand", "million", etc: "eleven hundred", "twenty-five thousand", "four million", "7 billion".
The reason for that is that, cognitively, it takes us longer to "figure out" a large number when expressed as one unit ("fifty-five thousand, five hundred and twenty-three") than when chunked ("five fifty-five twenty-three") (see, e.g., https://en.wikipedia.org/wiki/Chunking_(psychology))
How to express numbers with 6 or more digits is less well-defined.