Having tried everything but ask for help, I reckon that's the next step.
Jason suggested I install ZoneAlarm. I grabbed the free version and gave it a whirl. Much as I resent the need for such a program, it's implimentation is really quite elegant. I spent about ten minutes setting permissions for every piece of software Dad uses. With any luck, it will only pop up when something bad is happening.
I'm letting that computer sit for a while, and hoping for the worst. =)
What else can we fix?
The clock in my computer doesn't seem to move at anything close to realtime. I keep resetting it, and it keeps being wrong. This morning, it was off by 12 minutes. If you're a person who puts stuff off until the last minute, this kind of thing can really be a problem.
Went ahead and downloaded Atomic Clock Sync v2.0. More free software. This one checks once a day to make sure the clock's still accurate. I reckon that'll help.