Firstly, you are correct in your statement of what $PATH does. If you were to break it somehow (as per your third point), you will have to manually type in /usr/bin/xyz if you want to run a program in /usr/bin from the terminal. Depending on how individual programs work, this might break some programs that invoke other ones, as they will expect to just be able to run ls or something.
So if you were to play around with $PATH, I would suggest saving it somewhere first. Use the command line instruction
echo $PATH > someRandomFile.txt
to save it in someRandomFile.txt
You can change $PATH using the export command. So
export PATH=someNewPath
HOWEVER, this will completely replace $PATH with someNewPath. Since items in path are separated by a ":", you can add items to it (best not to remove, see above) by executing
export PATH=$PATH:newPath
The fact that it is an environmental variable means that programs can find out its value, ie it is something that is set about the environment that the program is running in. Other environmental variables include things like the current directory and the address of the current proxy.