When I connect to the internet it is frequently having a System Shutdown where I get a 1 minute countdown to a rebooting and an error message stating system shutdown "initiated by NT AUTHORITY\SYSTEM. Windows must now restart because the Remote Procedure Call (RPC) service terminated unexpectedly." Also I cannot cancel the shutdown by typing "shutdown -a" in at the DOS prompt as when I try to bring the DOS command box up I get a message saying

16 bit MS-DOS Subsystem
C:\WINDOWS\System32\command.com
C:\WINDOWS\SYSTEM32\AUTOEXEC.NT. The system file is not suitable for running MS-DOS and Microsoft Windows applications. Choose 'Close' to terminate the application.

How can I fix one of both of these problems as it is currently very difficult to use the internet for any period of time?

Do you have access to another computer? If so download Stinger to a floppy disk and run it on the infected computer. Also after you get the computer disinfected enable the xp firewall and get all of the available critical updates for your computer.

Also see this article