Hey knutove, and thanks for the response. Midterms and such have kept me from posting here again sooner.
I had previously tried connecting the box directly to the cable modem a while back, but with mixed results. Seeing as this means disconnecting every other machine from the internet, I had to phone someone to check on the status of the webpage. They claimed it was down, but I think that was because of an unrelated DNS error... but I digress.
I took your advice, and again plugged the box straight into my cable modem, and lo and behold, the website worked. Putting it back behind the router, and it worked for a few minutes, then stopped working, then started again, etc. There was much gnashing of teeth. I happened to be together with a few geek buddies of mine over the weekend, and so we decided to sit down for a bit and put our heads together. Eventually, thanks to the God-send that is Google, we found mention of the particular brand of router I was using (a Linksys Network Everywhere router) having the exact same problems I had experienced - flaky and random behaviour regarding port 80. One review even mentioned that he had called and complained to Linksys, only to be told that they were aware of the problem, and were not planning to fix it. Harrumph.
Google will give you what you want, but a direct link to reviews on Amazon.com is here:
I hope that anyone who is going through the same headache I went through, can be helped by some of this. Moral of the story: You get what you pay for with a $14 router.