Is it better to just close the lid on a macbook pro or should I shut it down?
That depends on you; many computer users almost never shut down their computers. It's fine to put it to sleep if you know you are going use the computer again soon. Sometimes though, it's best to shut it down if you're not going to use it for a day or two, as it will save a little bit of energy and heat from being wasted in your laptop. Also, shutting down completely wipes away the computer's RAM, which might have some corrupt/junk data left in it from various things the computer does. It also lets the computer's components cool down. I definitely recommend you shutting down at least once a month or so.
When I restart a computer, and I have multiple desktops open, will the desktops all restart with settings saved to reopen applications on those same desktops?
From Os X 10.8 and later, yes they will, all your desktops will survive reboots and all the applications will "remember" in which desktop they were open. Though, when you hit 'restart' you will be prompted a message to confirm, and there will be this option: Reopen windows when loging back in, make sure it's checked.
If I have an application open on desktop 4 and options --> this desktop so that the application only opens on desktop 4, will desktops 2 and 3 automatically open if I did not have them open at restart?
No, they won't. As I said earlier, if you have desktop 1-2-3-4 open and then you restart the system, they will reappear then. If you close a desktop, for example 2, desktop 3 will become desktop 2 and desktop 4 will become desktop 3.
So if you have only two desktops open and you open the app that was saved to work on desktop 4, former desktop 2 and 3 won't appear, the app will just bind on the new desktop where you open it.
At restart, if I just open the application that is saved on desktop 4, will applications 2 and 3 restart automatically?
As stated above, they won't.