What is the purpose of PATH?
It is how your shell finds programs. When you type ls, for example, that is running a program called ls which lives in /bin on most systems, including Mac OS X. Your shell can only find that because /bin is in the PATH.
The shell searches for programs in PATH order, left-to-right. If there are two ls programs on your system, and their directories are both in your PATH, it will find the one in the directory listed first in your PATH.
What is the difference between /etc/paths and ~/.bash_profile
Paths set in /etc/paths are added to the PATH at startup in all shells on Mac OS X.
~/.bash_profile is just one of several shell scripts run by Bash at startup, and several of these are Bash-specific. Therefore, if you change your shell — Mac OS X also ships tcsh and zsh, for example — changes made to /etc/bashrc don't apply to your new shell.
Also, /etc/paths is a system-level configuration file, while ~/.bash_profile is per-user. If you had multiple interactive users set up on your Mac, you wouldn't want to add directories to /etc/paths that you didn't want to appear in everyone's PATH. The same is true for /etc/bashrc, except that of course that only applies to users that use Bash as their shell.
What is the difference between PATH and $PATH and ${PATH}?
You usually only see the variable without the sigil when you are setting the variable. FOO=bar sets the FOO environment variable to the string bar. If you type set, the shell shows you all environment variables, and in that case you also see the variable without any sigil.
$PATH and ${PATH} are usually interchangeable. They tell the shell to expand the current PATH variable's value in-place. The difference has to do with how and where you use them. For example:
$ FOO=bar
$ echo $FOO
bar
$ echo $FOOx
$ echo ${FOO}x
barx
In the second echo command, the shell prints nothing because it asks the shell to print a variable called FOOx, which doesn't exist; the shell treats nonexistent variables as empty. Because the third echo uses the curly brace syntax, it allows the shell to see that you are requesting the FOO variable, and that the x is just another character you want printed immediately afterward.
There's another way to get the same effect without curly braces, by the way:
$ echo "$FOO"x
barx
Environment variables are expanded in double-quotes, and the quotes separate the x from the variable expansion so the shell does the right thing.
I'm using the FOO environment variable here just for clarity. Everything I've written above applies just as well to PATH, since it is just another environment variable. The example texts would just be a lot longer.
What does export PATH mean? When do we export?
Taking the above FOO example, if you were to run another program, it wouldn't see the FOO variable because it is not exported. It lives only within that one shell instance:
$ echo $FOO
bar
$ bash
$ echo $FOO
$ exit
$ export FOO
$ bash
$ echo $FOO
bar
When I first run a new bash shell under the one I was already using, then try to show the value of FOO, I get a blank because FOO wasn't exported for sub-programs to see. Then I exited the second bash instance, exported it, ran bash again, and now the second instance sees the value of FOO.
You use export when you want sub-programs to see values set in the shell, and don't use it when you don't want that to happen.
Typically, I don't export temporary variables in shell scripts, because I don't want them to change how programs run from that shell script behave.
I'm not going to answer any of your other questions. You're only supposed to ask one question at a time. I've only answered this many because they're vaguely related. Everything else belongs in a separate question.
