Code three or code 3?

It all depends. For the sake of brevity and taking up less space on paper, it's customary to write Code3 etc. But most newspaper style books dictate that the numerals one through ten (1 through 10) be spelled out in letters, anything larger is to be typed in numerals. So it depends on what medium you're writing for and whether your object is clarity or convenience.

Yes, I learned that, too... however, you'd never write out nine-one-one, and I think with a Code situation, you'd write Code 3, Code 5, etc., regardless of the genre.

Safety and standard communications codes are written as the number not the text. This is because that is the way they are officially denoted. For example:

Official Ten-Code List Association of Public Communications Officers (APCO)
10-0 Caution
10-1 Unable to copy -- change location
10-2 Signal good
10-3 Stop transmitting
10-4 Acknowledgement (OK)
10-5 Relay
10-6 Busy -- stand by unless urgent
10-7 Out of service
10-8 In service
10-9 Repeat
10-10 Fight in progress

So in using the APA standard, for example, you would write it exactly as it is in the defining standard. In terms of practicality, could you imagine the weird list you would get if you listed these as written in text rather than as numbers. 10-0 is now last in the list as it would be written ten-zero.

The same for the code-3 code-4. If written with the number in the standard then that is the way you also write it; by APA.