Some Linux distributions have the ability to be installed over a network. However, it should be noted that this is an endeavor that is intermediate to expert in difficulty.
You could use a usb flash drive.
You can install Windows or Linux using a USB memory stick.
You can't, you need the CD.
Any USB CD / DVD drive can easily be installed; just plug it into one of the USB ports on the Eee PC. Both Windows XP and Linux should be able to see it; and you can boot from it in the BIOS.
It is a CD the has some form of Linux, an operating system for your computer (Microsoft Windows is an operating system). There are CDs of Linux that are just used to install Linux, or there are LiveCDs that you just put in and reboot. Linux will be running without effecting the rest of the computer.
Most Linux distributions including CD and USB do support wireless adapters. The problem is that not all adapters are supported.
A program can be installed off a CD or DVD, or a flash drive.
Yes, although no modern distributions directly support system installs from them. You can still use a floppy disk to install Linux from a PXE boot server, from a CD that is incapable of booting from the BIOS, or from a USB Flash drive.
Not much. Depending on your BIOS, you can look at computer information, run hardware diagnostics and modify BIOS settings. You can boot from connected devices (Floppy, CD/DVD, USB etc.) to install windows or linux, or run pen-drive linux or utilities like memtest86+.
from cd, dvd, or thumbdrive
If you have a USB drive and your BIOS supports booting from a USB device then try that.
You could install it on an external drive without shutting down. You would download the live CD ISO and install is in a virtual machine (VirtualBox), add the external drive in the USB menu, and install as usual.