Shell – what does the -z flag mean here

You can always check the man page of your shell. man bash says:

Special Parameters
   #      Expands to the number of positional parameters in decimal.

Therefore a shell script can check how many parameters are given with code like this:

if [ "$#" -eq 0 ]; then
  echo "you did not pass any parameter"
fi

[ is another name for test. All three of those lines are command lines that run test with some options.

In the first line, this is standard testing: [ 5 -gt 4 ] is the same as test 5 -gt 4.

In the second, the expression is omitted, which means to exit false (0 arguments: Exit false (1)).

For the third case you are, I guess, using GNU coreutils. In GNU test the help text contains this note:

NOTE: [ honors the --help and --version options, but test does not.
test treats each of those as it treats any other nonempty STRING.

This is a non-POSIX extension; the motivation seems to be that test is required to treat those arguments as strings like any other. [ is able to distinguish the option case from the string case by the presence of the closing ] bracket.

Note that your shell will likely provide its own [, and so you'll have to /bin/\[ to use this version.