<<The former asks me for a Windows XP Disk, but doesn't accept it when I use one, and the latter simply tells me that "Windows cannot complete the system check".>>
It's asking for a MS XP install CD...and it must reflect the same SP-level that is on the system, it will not run if the CD reflects an earlier level.
Example: I have SP3 installed on both my desktops...to run SFC, I must use a CD that includes SP3 in the XP install package. I accomplish this by slipstreaming/updating my original XP CD, using nLite.
BC has a guide you may be interested in, http://www.bleepingcomputer.com/tutorials/slipstreaming-windows-xp-to-create-bootable-cd/
<<...I immediately get a popup saying "The webpage you requested is not available offline. Would you like to connect?">>
IMO, this is routine...it happens to me on occasion. Overcome by hitting the button to try again, IME.
As for Symantec and Google Desktop...perhaps those are damaged and need to be uninstalled/reinstalled...hard to be definitive without the error messages being known.
IMO, none of these sound like malware issues.