The mystery flags 3 and 5 actually work on some calculators as that documentation suggests and on some it does not. The reason to have the documentation is basically for emulators that want to nearly perfectly emulate the calculator. Otherwise, they aren't often useful.
As for your question about nz, nz is not a flag. 'nz' and 'z' (for conditional operations like jr, jp, ret and call) refer to the state of the flag. For example, call z,_label will call the subroutine at _label if the z flag is set, whereas call nz,_label calls it if the z flag is reset (call if not z). The same works with the c flag. You can perform those conditionals using c or nc. The other flags have more special names. For example, 'm' refers to the sign flag being set (m=minus), 'p' refers to it being reset (p=positive), 'po' refers to the parity flag being set (parity odd) and 'pe' refers to the parity flag being reset (parity even).