Will do. Thanks. I really appreciate you guys looking at the log.
Well, I found system32/drivers/core.cache.dsk yesterday and deleted it.
I only thought to look there after my mother read the Spy Sweeper logs backwards. She read the log for June 1st thinking it all came back, not realizing the June 2nd logs were clean and the time stamps for June 1st were fine starting at 4pm, which is after we went into safe mode that day. But it's good we had a false alarm because we found something we weren't expecting. But even before then, the scans and logs were looking clean, which kind of makes me paranoid, thinking there's something else lurking about that I missed.
Also, I remembered the old Spy Sweeper log she e-mailed me for June 1st and looked in Registry Edit for HKEY_LOCAL_MACHINE\SYSTEM\ControlSet001 (002 or 003)\Services\core, which is what she had before, and found nothing. That's good.
I'm 99.9% sure everything is going OK now.