The PATH environment variable tells the shell where to look for a program when you type its name on the command line.
For example, to search for files, you'll often use the program "find". "find" is just a regular program like any other, and it generally lives in "/usr/bin/". So that you don't have to manually type out "/usr/bin/find" every time you want to run it, you can instead add "/usr/bin/" to your PATH. Once "/usr/bin/" is in your PATH, you can just type "find", and the shell will locate and run the program.
The rest sounds like a homework problem, so I'll forego answering them for you.