Bash – Difference between quoting variables in shell script “if” statements

Exported variables are carried into the environment of commands executed by the shell that exported them, while non-exported variables are local to the current shell invocation. From the export man page:

The shell shall give the export attribute to the variables corresponding to the specified names, which shall cause them to be in the environment of subsequently executed commands.

set outputs the current environment, which includes any local non-exported variables. env is used to launch programs in a new environment, and with no arguments will output what that new environment would be. Since env is creating a new environment, only exported variables are brought through, as is the case for any program launched from that shell. For example, spawning a second shell within the first (I used $$ to represent prompts in the inner shell):

$ FOO=BAR
$ bash
$$ echo $FOO             # Note the empty line

$$ exit
$ export FOO
$ bash
$$ echo $FOO
BAR
$$

Note that it's the variable that's exported, not just its value. This means that once you export FOO, FOO becomes a global variable and shows up in subsequent environments, even if changed later:

$ export FOO
$ FOO=BAR
$ bash
$$ echo $FOO
BAR
$$

For one, --help is not a command, it is an argument that is often given to a command to get help using it. Meanwhile, man is a command, short for "manual". Manual pages are installed by many programs, and are a common way to find help about commands, as well as system calls, (e.g. fork()).

If a program installs a manual page, it can always be accessed via the man command, whereas --help is just a common convention, but need not be enforced—it could be just (and only) -h.

man also typically uses a pager, such as less, automatically, which can make viewing and searching through the information much easier.

Finally, you mention Bash programming in your question—none of this is unique to Bash. Bash doesn't care about the commands themselves or their arguments for the most part.