Please don't let this discourage you from trying Linux based systems, but I do not approve of your reason and will remind you that your actions taken to attempt to work around established IT policies can easily backfire. With that said you have had some good questions that I will answer in hopes that you will learn something and use it for a positive project.
You are correct that windows based applications generally do not run well under Linux based systems if at all, that is because the binaries and dependent libraries are different between the OSs. There are applications like wine or crossover that emulate a windows environment but the experience on various applications differ.
You can install some distros to USB drives, but they are generally in a persistent state that will not save any files or changes. The performance on a usb medium is lagging because it is being read from a limited medium and must store more in memory. You could look at slax which allows you to choose the specific packages and build a custom usb or livecd based installation, this would be the best option for simplicity.
However, on the computers you are looking to run it from the bios may be locked stopping you from booting from an external device such as cd or usb. So all efforts to accomplish this task may be in vain for that location, but it can still be used on personal computers such as your own computer or family members computers.
In addition, the facility will most likely have tools on the network that are monitoring for unknown patterns such as an unapproved OS or the applications on it, so the act of booting a Linux based system may be enough for them to track you.