Hopefully, some of the web apps for Firefox 16 beta also have offline support for doing stuff like typing notes, recording video, or playing games from within a web browser.
I read that Firefox 16 will use less RAM when it is release, so the extra RAM that the web apps uses won't be as noticeable depending on which apps you have installed.
I think web apps only run when you use them by clicking on the shortcut on your browser bookmark bar/app bar like regular programs which run when you click on the shortcut icon, on the startmenu and desktop, so they shouldn't use much memory unless you are a bunch of apps which are always updating like Facebook, Twitter, or chat programs.
A lot of web apps like Google Docs, Gmail, etc are a website loaded on a web browser since web apps are like websites, but instead of displaying text, ads, and pictures, web apps are used for gaming, Office work, social networking, and doing other tasks. I'm guesing they use as much memory as a modern website loaded on a new Tab or window.
Same for me, I wonder when it will be release since Firefox 15 is out now, and would FF Web apps would make more people upgrade their older version of Firefox to FF16, or stop using IE, Opera, Chrome, Maxthon ,etcto use Firefox web apps.
I got a feeling a lot of people will most likely install lots of game apps on Firefox to secretly play games in another tab/window when they are at school, work, or somewhere they should not be gaming.
There are USB versions of Firefox which let you run the Windows version of Firefox from a USB Flash drive which you can download at http://portableapps.com/apps/internet/firefox_portable if your school PC has USB, you can run the latest version of Firefox on your USB.
I can't say this for sure, but the PC permissions should be set up so you can only run programs what is on the desktop. But I've never taken a program and ran it on a PC.